2 Hessenberg varieties

In this section, we recall the definitions of Hessenberg varieties in type $A_{n-1}$ and their defining equations. We use the notation throughout this paper.

Let n be a positive integer. A Hessenberg function is a function $h: [n] \to [n]$ satisfying the following two conditions

1. 1. $h(1) \leq h(2) \leq \cdots \leq h(n)$ ;

2. 2. $h(j) \geq j$ for all $j \in [n]$ .

Note that $h(n)=n$ by definition. We frequently denote a Hessenberg function by listing its values in sequence, namely $h = (h(1), h(2), \ldots , h(n))$ . It is useful to see a Hessenberg function h pictorially by drawing a configuration of boxes on a square grid of size $n \times n$ whose shaded boxes consist of boxes in the i-th row and the j-th column such that $i \leq h(j)$ for $i, j \in [n]$ .

Example 2.1. Let $n=5$ . Then, $h=(3,3,4,5,5)$ is a Hessenberg function and the configuration of the shaded boxes is shown in Figure 1.

Figure 1 The configuration corresponding to $h=(3,3,4,5,5)$

Let $\mathfrak {gl}_n({\mathbb {C}})$ be the set of $n \times n$ matrices. For a Hessenberg function h, we define

(2.1)

which is called the Hessenberg space associated to h.

The (full) flag variety $Fl({\mathbb {C}}^n)$ is the set of nested complex linear subspaces of ${\mathbb {C}}^n$ :

Let B be the set of upper triangular matrices in the general linear group $\operatorname {\mathrm {GL}}_n({\mathbb {C}})$ . As is well known, the flag variety $Fl({\mathbb {C}}^n)$ can be identified with $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ . Indeed, each flag $V_\bullet \in Fl({\mathbb {C}}^n)$ is determined by a matrix g whose first j column vectors generate the j-th vector space $V_j$ for $j \in [n]$ . The correspondence $gB \mapsto V_{\bullet }$ above gives the identification $Fl({\mathbb {C}}^n) \cong \operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ . For a linear operator $A: {\mathbb {C}}^n \rightarrow {\mathbb {C}}^n$ and a Hessenberg function $h: [n] \to [n]$ , the Hessenberg variety $\operatorname {\mathrm {Hess}}(A,h)$ is defined to be the following subvariety of the flag variety $Fl({\mathbb {C}}^n)$ :

(2.2)

Hessenberg varieties are introduced in [Reference De Mari, Procesi and Shayman11, Reference De Mari and Shayman12]. Note that if $h=(n,n,\ldots ,n)$ , then $\operatorname {\mathrm {Hess}}(A,h)=Fl({\mathbb {C}}^n)$ . By identifying $Fl({\mathbb {C}}^n)$ with $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ , we can write the definition above as

(2.3) $$ \begin{align} \operatorname{\mathrm{Hess}}(A,h) = \{ gB \in \operatorname{\mathrm{GL}}_n({\mathbb{C}})/B \mid g^{-1} A g \in H(h) \}, \end{align} $$

where $H(h)$ is the Hessenberg space defined in equation (2.1).

Definition 2.2. Let $A \in \mathfrak {gl}_n({\mathbb {C}})$ and $i,j \in [n]$ . We define a polynomial $F_{i,j}^{A}$ on $\operatorname {\mathrm {GL}}_n({\mathbb {C}})$ by

where $g=(v_1 \, \ldots \, v_n) \in \operatorname {\mathrm {GL}}_n({\mathbb {C}})$ is the decomposition into column vectors. In other words, the polynomial $F_{i,j}^{A}(g)$ is the determinant of the matrix obtained from g by replacing the i-th column vector of g to the j-th column vector of $A g$ .

Lemma 2.3. Let $A \in \mathfrak {gl}_n({\mathbb {C}})$ . For $g \in \operatorname {\mathrm {GL}}_n({\mathbb {C}})$ and $i,j \in [n]$ , we have

$$ \begin{align*} \frac{1}{\det(g)}F_{i,j}^{A}(g) = (g^{-1} A g)_{ij}, \end{align*} $$

where $(g^{-1} A g)_{ij}$ denotes the $(i,j)$ -th entry of the matrix $g^{-1} A g$ . In particular, we have

$$ \begin{align*} \operatorname{\mathrm{Hess}}(A,h) =\{ gB \in \operatorname{\mathrm{GL}}_n({\mathbb{C}})/B \mid F_{i,j}^{A}(g)=0 \ \text{for all} \ 1 \leq j \leq n-1 \ \text{and} \ h(j) < i \leq n \}. \end{align*} $$

Proof. Let $c_{ij}$ be the $(i,j)$ cofactor of g, namely $c_{ij}$ is obtained by multiplying the $(i,j)$ minor of g by $(-1)^{i+j}$ . Set $\tilde g=(c_{ij})_{i,j \in [n]}^t=(c_{ji})_{i,j \in [n]}$ . Since $g^{-1}=\frac {1}{\det (g)} \, \tilde g$ , it suffices to show that

(2.4) $$ \begin{align} F_{i,j}^{A}(g) = (\tilde g A g)_{ij}. \end{align} $$

We write $g=(v_1 \, \ldots \, v_n)$ and $A v_j=(b_{1j},\ldots ,b_{nj})^t$ . By the definition of $\tilde g$ , we have

$$ \begin{align*} (\tilde g \cdot A g)_{ij} = \sum_{k=1}^n c_{ki} b_{kj} = \det (v_1 \, \ldots \, v_{i-1} \ A v_j \ v_{i+1} \,\ldots \, v_n), \end{align*} $$

where we used the cofactor expansion along the i-th column for the last equality. This yields equation (2.4) as desired. The latter statement follows from the former statement and equation (2.3).

3 Regular nilpotent Hessenberg varieties

In this section, we review geometric properties for regular nilpotent Hessenberg varieties.

Let N be a regular nilpotent element in $\mathfrak {gl}_n({\mathbb {C}})$ , that is, a matrix whose Jordan canonical form consists of exactly one Jordan block with corresponding eigenvalue equal to $0$ . For a Hessenberg function h, $\operatorname {\mathrm {Hess}}(N,h)$ is called the regular nilpotent Hessenberg variety. If $h=(2,3,4,\ldots ,n,n)$ , then is called the Peterson variety, which is an object of an intense study by Dale Peterson [Reference Peterson31]. Surprisingly, the Peterson variety is related with the quantum cohomology of the flag variety ([Reference Kostant29, Reference Rietsch32]). We will explain the relation in Section 4. For any $p \in \operatorname {\mathrm {GL}}_n({\mathbb {C}})$ , one can see that $\operatorname {\mathrm {Hess}}(N,h) \cong \operatorname {\mathrm {Hess}}(p^{-1}Np,h)$ which sends $gB$ to $p^{-1}gB$ . Thus, we may assume that N is in Jordan form:

(3.1) $$ \begin{align} N = \begin{pmatrix} 0 & 1 & & \\ & \ddots & \ddots & \\ & & 0 & 1 \\ & & & 0 \\ \end{pmatrix}. \end{align} $$

We summarize geometric properties of $\operatorname {\mathrm {Hess}}(N,h)$ as follows.

As a property of regular nilpotent Hessenberg varieties, a special case of $\operatorname {\mathrm {Hess}}(N,h)$ can be decomposed into a product of smaller regular nilpotent Hessenberg varieties, as explained below. To explain this, we first recall the following terminology from [Reference Drellich13, Definition 4.4].

If h is a decomposable Hessenberg function, that is, $h(j)=j$ for some $j \in [n-1]$ , then the Hessenberg function h can be decomposed into two smaller Hessenberg functions $h_1$ and $h_2$ defined by $h_1=(h(1),\ldots ,h(j))$ and $h_2=(h(j+1)-j,\ldots ,h(n)-j)$ as shown in Figure 2.

Figure 2 The decomposition of $h=(2,3,3,5,5)$ into $h_1=(2,3,3)$ and $h_2=(2,2)$

Then, every $V_\bullet \in \operatorname {\mathrm {Hess}}(N,h)$ has $V_j = \operatorname {\mathrm {span}}_{\mathbb {C}} \{e_1,\ldots ,e_j\}$ , where $e_1,\ldots , e_n$ denote the standard basis of ${\mathbb {C}}^n$ and hence we have

(3.2) $$ \begin{align} \operatorname{\mathrm{Hess}}(N,h) \cong \operatorname{\mathrm{Hess}}(N_1, h_1) \times \operatorname{\mathrm{Hess}}(N_2, h_2), \end{align} $$

where $N_1$ and $N_2$ are the regular nilpotent matrices in Jordan canonical form of size j and $n-j$ , respectively ([Reference Drellich13, Theorem 4.5]).

In a recent paper, Abe–Insko gave the condition that $\operatorname {\mathrm {Hess}}(N,h)$ is normal as follows.

There is a natural partial order on Hessenberg functions. For two Hessenberg functions $h:[n] \rightarrow [n]$ and $h':[n] \rightarrow [n]$ , we say $h' \subset h$ if $h'(j) \leq h(j)$ for all $j \in [n]$ . Note that if $h' \subset h$ , then $\operatorname {\mathrm {Hess}}(N,h')$ is a closed subvariety of $\operatorname {\mathrm {Hess}}(N,h)$ by the definition (2.2). Since $h=(2,3,4,\ldots ,n,n)$ is the minimal Hessenberg function among indecomposable Hessenberg functions with respect to the partial order $\subset $ , the Peterson variety $\operatorname {\mathrm {Pet}}_n$ is the minimal Hessenberg variety among indecomposable regular nilpotent Hessenberg varieties with respect to the inclusion.

Let h be a Hessenberg function. One can see that the Hessenberg space $H(h)$ in equation (2.1) is stable under the adjoint action of B, so this induces a B-action on the quotient space $\mathfrak {gl}_n({\mathbb {C}})/H(h)$ . We denote by $\overline {x}$ the image of $x \in \mathfrak {gl}_n({\mathbb {C}})$ under the surjection $\mathfrak {gl}_n({\mathbb {C}}) \rightarrow \mathfrak {gl}_n({\mathbb {C}})/H(h)$ . In [Reference Abe, DeDieu, Galetto and Harada1], we consider the vector bundle $\operatorname {\mathrm {GL}}_n({\mathbb {C}}) \times _B (\mathfrak {gl}_n({\mathbb {C}})/H(h))$ over the flag variety $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ and its section $s_A$ for $A \in \mathfrak {gl}_n({\mathbb {C}})$ defined by

$$ \begin{align*} s_A: \operatorname{\mathrm{GL}}_n({\mathbb{C}})/B \rightarrow \operatorname{\mathrm{GL}}_n({\mathbb{C}}) \times_B (\mathfrak{gl}_n({\mathbb{C}})/H(h)); \ gB \mapsto [g,\overline{g^{-1} A g}], \end{align*} $$

where we denote by $[g,\overline {x}]$ the image of $(g,\overline {x}) \in \operatorname {\mathrm {GL}}_n({\mathbb {C}}) \times (\mathfrak {gl}_n({\mathbb {C}})/H(h))$ under the surjection $\operatorname {\mathrm {GL}}_n({\mathbb {C}}) \times (\mathfrak {gl}_n({\mathbb {C}})/H(h)) \rightarrow \operatorname {\mathrm {GL}}_n({\mathbb {C}}) \times _B (\mathfrak {gl}_n({\mathbb {C}})/H(h))$ such that $[g,\overline {x}]=[gb,\overline {b^{-1}xb}]$ for all $b \in B$ . By the definition (2.3), one can see that the zero set of $s_A$ is the Hessenberg variety $\operatorname {\mathrm {Hess}}(A,h)$ .

In general, let $\pi :E \rightarrow X$ be a vector bundle of rank r over a scheme X. If s is a section of E, then the zero scheme $\mathcal {Z}(s)$ of s is defined as follows (cf. [Reference Fulton20]). Let $(U_i, \varphi _i)_i$ be a local trivialization of E, that is, an open covering $\{U_i\}_i$ of X and isomorphisms $\varphi _i$ of $\pi ^{-1}(U_i)$ with $U_i \times {\mathbb {C}}^r$ over $U_i$ such that the composites $\varphi _i \circ \varphi _j^{-1}$ are linear. Let $s_i: U_i \rightarrow {\mathbb {C}}^r$ determine the section s on $U_i$ , $s_i=(s_{i_1},\ldots ,s_{i_r})$ , $s_{i_m}$ in the coordinate ring of $U_i$ ; then $\mathcal {Z}(s)$ is defined in $U_i$ by the ideal generated by $s_{i_1},\ldots ,s_{i_r}$ .

Motivated by the discussion above, we use the following definition introduced in [Reference Abe, DeDieu, Galetto and Harada1].

Locally around $gB \in \operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ , the section $s_N$ is represented by a collection of regular functions, and the scheme $\mathcal {Z}(N, h)$ is locally the zero scheme of those functions. (See [Reference Abe, DeDieu, Galetto and Harada1, Lemma 3.4] for the detail.)

We analyze the intersection of $\operatorname {\mathrm {Hess}}(N,h)$ and the opposite Schubert cell $\Omega _e^\circ $ associated with the identity element e. Recall that $\Omega _e^\circ $ is the $B^{-}$ -orbit of the identity flag $eB/B$ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ , where $B^{-}$ denotes the set of lower triangular matrices in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})$ . In particular, each flag $V_\bullet \in \Omega _e^\circ $ has $V_j$ spanned by the first j columns of a matrix with $1$ ’s in the diagonal positions and $0$ ’s to the right of these $1$ ’s (cf. [Reference Fulton19, Section 10.2]). Thus, one can see that the opposite Schubert cell $\Omega _e^\circ $ can be regarded as the following affine space:

$$ \begin{align*} \Omega_e^\circ \cong \left\{ \left. \left( \begin{array}{@{\,}ccccc@{\,}} 1 & & & & \\ x_{21} & 1 & & & \\ x_{31} & x_{32} & 1 & & \\ \vdots& \vdots & \ddots & \ddots & \\ x_{n1} & x_{n2} & \cdots & x_{n \, n-1} & 1 \end{array} \right) \ \right| x_{ij} \in {\mathbb{C}} \ (1 \leq j < i \leq n) \right\} \cong {\mathbb{C}}^{\frac{1}{2}n(n-1)}. \end{align*} $$

Note that $\Omega _e^\circ $ is an affine open set of the flag variety $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ and its coordinate ring is isomorphic to the polynomial ring ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]$ . Thus, we may identify $\Omega _e^\circ $ as $\operatorname {\mathrm {Spec}} {\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]$ . From now on, we write

for simplicity. Since the first column vector of $Ng$ is $(x_{21},x_{31},x_{41},\ldots ,x_{n1},0)^t$ and the j-th column vector of $Ng$ is $(\underbrace {0,\ldots ,0}_{j-2},\underbrace {1,x_{j+1 \, j},\ldots ,x_{nj},0}_{n-j+2})^t$ for $j \geq 2$ , we can explicitly write

(3.3) $$ \begin{align} F_{i,1} &= \left| \begin{array}{@{\,}ccccc@{\,}} 1 & 0 & \cdots & 0 & x_{21} \\ x_{21} & 1 & \ddots & \vdots & x_{31} \\ \vdots & x_{32} & \ddots & 0 & \vdots \\ \vdots& \vdots & \ddots & 1 & \vdots \\ x_{i1} & x_{i2} & \cdots & x_{i \, i-1} & x_{i+1 \, 1} \end{array} \right| \ \ \ \text{for} \ j=1; \end{align} $$

(3.4) $$ \begin{align} F_{i,j} &= \left| \begin{array}{@{\,}ccccc@{\,}} 1 & 0 & \cdots & 0 & 1 \\ x_{j\,j-1} & 1 & \ddots & \vdots & x_{j+1\,j} \\ x_{j+1\,j-1} & x_{j+1\,j} & \ddots & 0 & \vdots \\ \vdots& \vdots & \ddots & 1 & \vdots \\ x_{i\,j-1} & x_{ij} & \cdots & x_{i \, i-1} & x_{i+1 \, j} \end{array} \right| \ \ \ \text{for} \ j \geq 2 \end{align} $$

from Definition 2.2. Here, we take the convention that $x_{n+1\,j}=0$ . The determinant of $g \in \Omega _e^\circ $ is $1$ , so we have

(3.5) $$ \begin{align} F_{i,j} = F_{i,j}^{N}(g) = (g^{-1}Ng)_{ij} \ \ \ \text{for} \ g \in \Omega_e^\circ \end{align} $$

by Lemma 2.3. We set

(3.6)

By the discussion above, we have the following.

Proposition 3.6. Let h be a Hessenberg function. Then

$$ \begin{align*} \mathcal{Z}(N,h)_e = \operatorname{\mathrm{Spec}} {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text{and} \ h(j) < i \leq n). \end{align*} $$

In other words, the set of global sections $\Gamma (\mathcal {Z}(N,h)_e, \mathcal {O}_{\mathcal {Z}(N,h)_e})$ is given by

(3.7) $$ \begin{align} \Gamma(\mathcal{Z}(N,h)_e, \mathcal{O}_{\mathcal{Z}(N,h)_e}) \cong {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text{and} \ h(j) < i \leq n). \end{align} $$

In particular, if h is indecomposable, then the coordinate ring ${\mathbb {C}}[\operatorname {\mathrm {Hess}}(N,h) \cap \Omega _e^\circ ]$ of the open set $\operatorname {\mathrm {Hess}}(N,h) \cap \Omega _e^\circ $ in $\operatorname {\mathrm {Hess}}(N,h)$ is

(3.8) $$ \begin{align} {\mathbb{C}}[\operatorname{\mathrm{Hess}}(N,h) \cap \Omega_e^\circ] = {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-2] \ \text{and} \ h(j) < i \leq n) \end{align} $$

by Theorem 3.5.

4 The main theorem

It can be extracted from Peterson’s statements in [Reference Peterson31] that the coordinate ring ${\mathbb {C}}[\operatorname {\mathrm {Pet}}_n \cap \Omega _e^\circ ]$ is isomorphic to the quantum cohomology ring of the flag variety $QH^*(Fl({\mathbb {C}}^n))$ . This result is unpublished, so we also refer the reader to [Reference Kostant29, Reference Rietsch32] for the result. In this section, we first review an explicit presentation for the quantum cohomology ring $QH^*(Fl({\mathbb {C}}^n))$ given by [Reference Ciocan-Fontanine16, Reference Givental and Kim22]. Then we introduce a further quantization of the ring presentation for $QH^*(Fl({\mathbb {C}}^n))$ in natural way. Our main theorem relates $\Gamma (\mathcal {Z}(N,h)_e, \mathcal {O}_{\mathcal {Z}(N,h)_e})$ and our quantized ring presentation.

The quantum cohomology ring of the flag variety $QH^*(Fl({\mathbb {C}}^n))$ is given by

$$ \begin{align*} QH^*(Fl({\mathbb{C}}^n)) = {\mathbb{C}}[q_1,\ldots,q_{n-1}] \otimes_{\mathbb{C}} H^*(Fl({\mathbb{C}}^n)) \end{align*} $$

as ${\mathbb {C}}$ -vector spaces. Here, $q_1,\ldots ,q_{n-1}$ are called the quantum parameters. The product structure of $QH^*(Fl({\mathbb {C}}^n))$ is a certain deformation of the cup product in the ordinary cohomology $H^*(Fl({\mathbb {C}}^n))$ . In order to describe an explicit presentation for $QH^*(Fl({\mathbb {C}}^n))$ , we need quantized elementary symmetric polynomials. Consider the matrix

and define quantized elementary symmetric polynomials $\check E_1^{(n)}, \ldots , \check E_n^{(n)}$ in the polynomial ring ${\mathbb {C}}[x_1,\ldots ,x_n,q_1,\ldots ,q_{n-1}]$ by the following equation

$$ \begin{align*} \det(\lambda I_n - \check M_n) = \lambda^n - \check E_1^{(n)} \lambda^{n-1} + \check E_2^{(n)} \lambda^{n-2} + \cdots + (-1)^n \check E_n^{(n)}, \end{align*} $$

where $I_n$ is the identity matrix of size n. By using the cofactor expansion along the s-th column for $\det (\lambda I_s - \check M_s)$ , we have

$$ \begin{align*} \det(\lambda I_s - \check M_s) = (\lambda-x_s) \det(\lambda I_{s-1} - \check M_{s-1}) +q_{s-1} \det(\lambda I_{s-2} - \check M_{s-2}). \end{align*} $$

This implies the following recursive formula

$$ \begin{align*} \check E_r^{(s)} = \check E_r^{(s-1)} + \check E_{r-1}^{(s-1)} x_s + \check E_{r-2}^{(s-2)} q_{s-1} \ \ \ \text{for} \ 1 \leq r \leq s \leq n, \end{align*} $$

where we take the convention that $\check E_0^{(s-1)}=1, \check E_{-1}^{(s-2)}=0$ whenever $r=1$ and $\check E_{s}^{(s-1)}=0$ whenever $r=s$ . By setting $q_s = 0$ for all $s \in [n-1]$ , $\check E_i^{(n)}$ is the (ordinary) i-th elementary symmetric polynomial in the variables $x_1,\ldots ,x_n$ .

Theorem 4.1 [Reference Ciocan-Fontanine16, Reference Givental and Kim22].

There is an isomorphism of ${\mathbb {C}}$ -algebras

(4.1) $$ \begin{align} QH^*(Fl({\mathbb{C}}^n)) \cong {\mathbb{C}}[x_1,\ldots,x_n,q_1,\ldots,q_{n-1}]/(\check E_1^{(n)}, \ldots, \check E_n^{(n)}). \end{align} $$

Theorem 4.4 (D. Peterson and [Reference Kostant29, Reference Rietsch32]).

There is an isomorphism of ${\mathbb {C}}$ -algebras

$$ \begin{align*} {\mathbb{C}}[\operatorname{\mathrm{Pet}}_n \cap \Omega_e^\circ] \cong QH^*(Fl({\mathbb{C}}^n)). \end{align*} $$

By Theorems 4.1 and 4.4, we obtain an isomorphism

(4.2) $$ \begin{align} {\mathbb{C}}[\operatorname{\mathrm{Pet}}_n \cap \Omega_e^\circ] \cong {\mathbb{C}}[x_1,\ldots,x_n,q_1,\ldots,q_{n-1}]/(\check E_1^{(n)}, \ldots, \check E_n^{(n)}) \end{align} $$

as ${\mathbb {C}}$ -algebras. One can see from [Reference Rietsch32, Theorems 3.3 and 4.2] that the isomorphism above sends $x_{ij}$ to $\check E_{i-j}^{(n-j)}$ under the presentation (3.8) for $h=(2,3,4,\ldots ,n,n)$ . We generalize equation (4.2) to arbitrary Hessenberg function h.

Definition 4.6. Consider the matrix

and define $q_{rs}$ -quantized elementary symmetric polynomials $E_1^{(n)}, \ldots , E_n^{(n)}$ in the polynomial ring ${\mathbb {C}}[x_1,\ldots ,x_n,q_{rs} \mid 1 \leq r < s \leq n]$ by the following equation

$$ \begin{align*} \det(\lambda I_n - M_n) = \lambda^n - E_1^{(n)} \lambda^{n-1} + E_2^{(n)} \lambda^{n-2} + \cdots + (-1)^n E_n^{(n)}. \end{align*} $$

In other words, $E_i^{(n)}$ is the coefficient of $\lambda ^{n-i}$ for $\det (\lambda I_n - M_n)$ multiplied by $(-1)^i$ .

Lemma 4.8. For $1 \leq r \leq s \leq n$ , we have

$$ \begin{align*} E_r^{(s)} = E_r^{(s-1)} + E_{r-1}^{(s-1)} x_s + \sum_{k=1}^{r-1} E_{r-1-k}^{(s-1-k)} q_{s-k \, s} \end{align*} $$

with the convention that $E_0^{(*)}=1$ for arbitrary $*$ , $\sum _{k=1}^{r-1} E_{r-1-k}^{(s-1-k)} q_{s-k \, s}=0$ whenever $r=1$ , and $E_{s}^{(s-1)}=0$ whenever $r=s$ .

Proof. It follows from the cofactor expansion along the s-th column for $\det (\lambda I_s - M_s)$ that $\det (\lambda I_s - M_s)$ is equal to

(4.3) $$ \begin{align} &(-1)^{s+2} q_{1s} + (-1)^{s+3} q_{2s} \det(\lambda I_1 - M_1) + \cdots + (-1)^{s+s} q_{s-1 \, s} \det(\lambda I_{s-2} - M_{s-2}) \\ & \hspace{20pt} + (-1)^{s+s} (\lambda - x_s) \det(\lambda I_{s-1} - M_{s-1}). \notag \end{align} $$

Since we have

$$ \begin{align*} \det(\lambda I_k - M_k) = \lambda^k - E_1^{(k)} \lambda^{k-1} + E_2^{(k)} \lambda^{k-2} + \cdots + (-1)^k E_k^{(k)} \end{align*} $$

for each $1 \leq k \leq s-1$ by the definition, the coefficient of $\lambda ^{s-r}$ for equation (4.3) is given by

$$ \begin{align*} &(-1)^r q_{s-r+1 \, s} + (-1)^{r} q_{s-r+2 \, s} E_1^{(s-r+1)} + \cdots + (-1)^{r} q_{s-1 \, s} E_{r-2}^{(s-2)} \\ & \hspace{20pt} + (-x_s) (-1)^{r-1} E_{r-1}^{(s-1)} + (-1)^{r} E_r^{(s-1)}. \end{align*} $$

Namely, the coefficient of $\lambda ^{s-r}$ for $\det (\lambda I_s - M_s)$ is

$$ \begin{align*} (-1)^r \big(q_{s-r+1 \, s} + q_{s-r+2 \, s} E_1^{(s-r+1)} + \cdots + q_{s-1 \, s} E_{r-2}^{(s-2)} + x_s E_{r-1}^{(s-1)} + E_r^{(s-1)} \big), \end{align*} $$

as desired.

Example 4.9. Let $n=3$ . Then the $E_r^{(s)}$ have the following form.

$$ \begin{align*} &E_0^{(1)} = 1, \ \ \ E_1^{(1)} = E_0^{(0)}x_1 =x_1, \\ &E_0^{(2)} = 1, \ \ \ E_1^{(2)} = E_1^{(1)} + E_0^{(1)}x_2 = x_1+x_2, \ \ \ E_2^{(2)} = E_1^{(1)}x_2 + E_0^{(0)}q_{12} = x_1x_2+q_{12}, \\ &E_0^{(3)} = 1, \ \ \ E_1^{(3)} = E_1^{(2)} + E_0^{(2)}x_3 = x_1+x_2+x_3, \\ &E_2^{(3)} = E_2^{(2)} + E_{1}^{(2)}x_3 + E_0^{(1)}q_{23} = x_1x_2+x_1x_3+x_2x_3+q_{12}+q_{23}, \\ &E_3^{(3)} = E_{2}^{(2)}x_3 + E_1^{(1)}q_{23} + E_0^{(0)}q_{13}= x_1x_2x_3+x_1q_{23}+x_3q_{12}+q_{13}. \end{align*} $$

Definition 4.10. Let $h: [n] \to [n]$ be a Hessenberg function. For each $1 \leq i \leq j \leq n$ , we define $^h E_i^{(j)}$ as the polynomial $E_i^{(j)}$ by setting $q_{rs}=0$ for all $2 \leq s \leq n$ and $1 \leq r \leq n-h(n+1-s)$ :

The surviving variables in the polynomial $^h E_i^{(j)}$ are pictorially shown as follows. Let $w_0$ be the longest element of the symmetric group $\mathfrak S_n$ on n letters $[n]$ . Consider the matrix

(4.4) $$ \begin{align} w_0 M_n w_0 = \left( \begin{array}{@{\,}ccccc@{\,}} x_n & -1 & 0 & \cdots & 0 \\ q_{n-1 \, n} & x_{n-1} & -1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ q_{2n} & \cdots & q_{23} & x_2 & -1 \\ q_{1n} & \cdots & q_{13} & q_{12} & x_1 \end{array} \right), \end{align} $$

where $w_0$ is regarded as the permutation matrix. Then $w_0 M_n w_0 \in H(h)$ if and only if $q_{rs}=0$ for all $2 \leq s \leq n$ and $1 \leq r \leq n-h(n+1-s)$ where $H(h)$ is the Hessenberg space defined in equation (2.1). In other words, the surviving variables in the polynomial $^h E_i^{(j)}$ are pictorially the variables $q_{rs}$ arranged in $w_0 M_n w_0$ such that $q_{rs}$ belongs to the configuration of the shaded boxes for the Hessenberg function h.

Example 4.11. Let $n=5$ and $h=(3,3,4,5,5)$ , which is depicted in Example 2.1. Then $^h E_i^{(j)}$ is a polynomial in the variables $x_1,\ldots ,x_5,q_{12},q_{23},q_{34},q_{35},q_{45}$ for $1 \leq i \leq j \leq 5$ , as shown in Figure 3.

Figure 3 The polynomial $^h E_i^{(j)} \in {\mathbb {C}}[x_1,\ldots ,x_5,q_{12},q_{23},q_{34},q_{35},q_{45}]$ for $h=(3,3,4,5,5)$

We now state the main theorem of this paper.

Theorem 4.13. Let $h: [n] \to [n]$ be a Hessenberg function and $\mathcal {Z}(N,h)_e$ the intersection defined in equation (3.6). Then there is an isomorphism of ${\mathbb {C}}$ -algebras

(4.5) $$ \begin{align} \Gamma(\mathcal{Z}(N,h)_e, \mathcal{O}_{\mathcal{Z}(N,h)_e}) \cong \frac{{\mathbb{C}}[x_1,\ldots,x_n, q_{rs} \mid 2 \leq s \leq n, n-h(n+1-s)<r<s]}{(^h E_1^{(n)}, \ldots, \, ^h E_n^{(n)})}, \end{align} $$

which sends $x_{ij}$ to $^h E_{i-j}^{(n-j)}$ under the presentation (3.7). In particular, if h is indecomposable, then there is an isomorphism of ${\mathbb {C}}$ -algebras

(4.6) $$ \begin{align} {\mathbb{C}}[\operatorname{\mathrm{Hess}}(N,h) \cap \Omega_e^\circ] \cong \frac{{\mathbb{C}}[x_1,\ldots,x_n, q_{rs} \mid 2 \leq s \leq n, n-h(n+1-s)<r<s]}{(^h E_1^{(n)}, \ldots, \, ^h E_n^{(n)})}, \end{align} $$

which sends $x_{ij}$ to $^h E_{i-j}^{(n-j)}$ under the presentation (3.8).

We will prove Theorem 4.13 in Section 7. For this purpose, one first see that the homomorphism in equation (4.5) is well defined and surjective in the next section.

5 Properties of $E_r^{(s)}$

In this section, we see relations between $x_s$ ’s, $q_{rs}$ ’s, and $E_r^{(s)}$ ’s. Then we construct an explicit map from $\Gamma (\mathcal {Z}(N,h)_e, \mathcal {O}_{\mathcal {Z}(N,h)_e})$ to our quotient ring ${\mathbb {C}}[x_1,\ldots ,x_n, q_{rs} \mid 2 \leq s \leq n, n-h(n+1-s)<r<s]/(^h E_1^{(n)}, \ldots , \, ^h E_n^{(n)})$ .

Lemma 5.1. For $1 \leq r < s \leq n$ , we have

$$ \begin{align*} q_{rs} = \left| \begin{array}{@{\,}cccccc@{\,}} 1 & 0 & \cdots & \cdots & 0 & E_{1}^{(s)}-E_{1}^{(s-1)} \\ E_1^{(s-1)} & 1 & 0 & & \vdots & E_{2}^{(s)}-E_{2}^{(s-1)} \\ E_2^{(s-1)} & E_1^{(s-2)} & 1 & \ddots & \vdots & E_{3}^{(s)}-E_{3}^{(s-1)} \\ \vdots & \ddots & \ddots & \ddots & 0 & \vdots \\ \vdots & & \ddots & E_1^{(r+1)} & 1 & E_{s-r}^{(s)}-E_{s-r}^{(s-1)} \\ E_{s-r}^{(s-1)} & \cdots & \cdots & E_2^{(r+1)} & E_1^{(r)} & E_{s-r+1}^{(s)}-E_{s-r+1}^{(s-1)} \end{array} \right| \end{align*} $$

in the polynomial ring ${\mathbb {C}}[x_1,\ldots ,x_n,q_{rs} \mid 1 \leq r < s \leq n]$ .

Proof. For $1 \leq s \leq n$ , it follows from Lemma 4.8 that

$$ \begin{align*} \begin{cases} E_1^{(s)} &= E_1^{(s-1)} + x_s \\ E_2^{(s)} &= E_2^{(s-1)} + E_{1}^{(s-1)} x_s + q_{s-1 \, s} \\ E_3^{(s)} &= E_3^{(s-1)} + E_{2}^{(s-1)} x_s + E_{1}^{(s-2)} q_{s-1 \, s} + q_{s-2 \, s} \\ &\vdots \\ E_{s-r+1}^{(s)} &= E_{s-r+1}^{(s-1)} + E_{s-r}^{(s-1)} x_s + E_{s-r-1}^{(s-2)} q_{s-1 \, s} + E_{s-r-2}^{(s-3)} q_{s-2 \, s} + \cdots + E_1^{(r)} q_{r+1 \, s} + q_{rs}. \end{cases} \end{align*} $$

In other words, we have

$$ \begin{align*} \left( \begin{array}{@{\,}cccccc@{\,}} 1 & & & & & \\ E_1^{(s-1)} & 1 & & & & \\ E_2^{(s-1)} & E_1^{(s-2)} & 1 & & & \\ \vdots & \ddots & \ddots & \ddots & & \\ \vdots & & \ddots & E_1^{(r+1)} & 1 & \\ E_{s-r}^{(s-1)} & \cdots & \cdots & E_2^{(r+1)} & E_1^{(r)} & 1 \end{array} \right) \left( \begin{array}{@{\,}c@{\,}} x_s \\ q_{s-1 \, s} \\ q_{s-2 \, s} \\ \vdots \\ q_{r+1 \, s} \\ q_{rs} \end{array} \right) = \left( \begin{array}{@{\,}c@{\,}} E_1^{(s)}-E_1^{(s-1)} \\ E_2^{(s)}-E_2^{(s-1)} \\ E_3^{(s)}-E_3^{(s-1)} \\ \vdots \\ E_{s-r}^{(s)}-E_{s-r}^{(s-1)} \\ E_{s-r+1}^{(s)}-E_{s-r+1}^{(s-1)} \\ \end{array} \right). \end{align*} $$

By Cramer’s rule, we obtain the desired result.

Set

(5.1)

and define a map $\varphi $ from ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]$ to the ring $Q_n$ by

(5.2) $$ \begin{align} \varphi: {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n] \to Q_n; \ x_{ij} \mapsto E_{i-j}^{(n-j)}. \end{align} $$

Proposition 5.2. The map $\varphi $ above sends

$$ \begin{align*} \varphi(x_{n-s+1 \, n-s} - x_{n-s+2 \, n-s+1}) &= x_s \ \ \ \text{for} \ s \in [n] \\ \varphi(-F_{n+1-r,n+1-s}) &= q_{rs} \ \ \ \text{for} \ 1 \leq r < s \leq n. \end{align*} $$

Here, we take the convention that $x_{n-s+2 \, n-s+1}=0$ whenever $s=1$ and $x_{n-s+1 \, n-s}=0$ whenever $s=n$ . In particular, $\varphi $ is surjective.

Proof. First, one can see from Definition 4.6 that

$$ \begin{align*} E_1^{(s)} =x_1+\dots+x_s \end{align*} $$

for any $s \in [n]$ , which implies that $x_s=E_1^{(s)} -E_1^{(s-1)}$ . By the definition of $\varphi $ we have $\varphi (x_{n-s+1 \, n-s} - x_{n-s+2 \, n-s+1}) = E_1^{(s)} -E_1^{(s-1)} = x_s$ for $s \in [n]$ . (Note that if $s=n$ , then we used the relation $E_1^{(n)}=0$ in the quotient ring $Q_n={\mathbb {C}}[x_1,\ldots ,x_n, q_{rs} \mid 1 \leq r < s \leq n]/(E_1^{(n)}, \ldots , E_n^{(n)})$ .)

Next, it follows from equation (3.3) that

(5.3) $$ \begin{align} \varphi(F_{n+1-r,1}) = \left| \begin{array}{@{\,}cccccc@{\,}} 1 & 0 & \cdots & \cdots & 0 & E_1^{(n-1)} \\ E_1^{(n-1)} & 1 & 0 & & \vdots & E_2^{(n-1)} \\ E_2^{(n-1)} & E_1^{(n-2)} & 1 & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 & \vdots \\ \vdots & & \ddots & E_1^{(r+1)} & 1 & E_{n-r}^{(n-1)} \\ E_{n-r}^{(n-1)} & \cdots & \cdots & E_2^{(r+1)} & E_1^{(r)} & E_{n-r+1}^{(n-1)} \end{array} \right| \end{align} $$

for $1 \leq r \leq n-1$ . On the other hand, by Lemma 5.1 we have

(5.4) $$ \begin{align} q_{rn} = \left| \begin{array}{@{\,}cccccc@{\,}} 1 & 0 & \cdots & \cdots & 0 & -E_{1}^{(n-1)} \\[2pt] E_1^{(n-1)} & 1 & 0 & & \vdots & -E_{2}^{(n-1)} \\ E_2^{(n-1)} & E_1^{(n-2)} & 1 & \ddots & \vdots & -E_{3}^{(n-1)} \\[2pt] \vdots & \ddots & \ddots & \ddots & 0 & \vdots \\[2pt] \vdots & & \ddots & E_1^{(r+1)} & 1 & -E_{n-r}^{(n-1)} \\[2pt] E_{n-r}^{(n-1)} & \cdots & \cdots & E_2^{(r+1)} & E_1^{(r)} & -E_{n-r+1}^{(n-1)} \end{array} \right| \end{align} $$

since $E_1^{(n)}=0, \ldots , E_{n-r+1}^{(n)}=0$ in the quotient ring $Q_n={\mathbb {C}}[x_1,\ldots ,x_n, q_{rs} \mid 1 \leq r < s \leq n]/(E_1^{(n)}, \ldots , E_n^{(n)})$ . By equations (5.3) and (5.4), we have $\varphi (F_{n-r+1,1})=-q_{rn}$ for $1 \leq r \leq n-1$ as desired.

If $s <n$ , then by equation (3.4) we have

$$ \begin{align*} \varphi(F_{n+1-r,n+1-s}) &= \left| \begin{array}{@{\,}cccccc@{\,}} 1 & 0 & \cdots & \cdots & 0 & 1 \\[2pt] E_1^{(s)} & 1 & 0 & & \vdots & E_1^{(s-1)} \\[2pt] E_2^{(s)} & E_1^{(s-1)} & 1 & \ddots & \vdots & E_2^{(s-1)} \\[2pt] \vdots & E_2^{(s-1)} & \ddots & \ddots & 0 & \vdots \\[2pt] \vdots & \vdots & \ddots & E_1^{(r+1)} & 1 & \vdots \\[2pt] E_{s-r+1}^{(s)} & E_{s-r}^{(s-1)} & \cdots & E_2^{(r+1)} & E_1^{(r)} & E_{s-r+1}^{(s-1)} \end{array} \right| \\[8pt] &= \left| \begin{array}{@{\,}cccccc@{\,}} 0 & 0 & \cdots & \cdots & 0 & 1 \\[2pt] E_1^{(s)}-E_1^{(s-1)} & 1 & 0 & & \vdots & E_1^{(s-1)} \\[2pt] E_2^{(s)}-E_2^{(s-1)} & E_1^{(s-1)} & 1 & \ddots & \vdots & E_2^{(s-1)} \\[2pt] \vdots & E_2^{(s-1)} & \ddots & \ddots & 0 & \vdots \\[2pt] \vdots & \vdots & \ddots & E_1^{(r+1)} & 1 & \vdots \\[2pt] E_{s-r+1}^{(s)}-E_{s-r+1}^{(s-1)} & E_{s-r}^{(s-1)} & \cdots & E_2^{(r+1)} & E_1^{(r)} & E_{s-r+1}^{(s-1)} \end{array} \right| \\[5pt] & \qquad \text{(by subtracting the last column from the first column)} \\[8pt] &= -\left| \begin{array}{@{\,}cccccc@{\,}} 1 & 0 & \cdots & \cdots & 0 & 0 \\[2pt] E_1^{(s-1)} & 1 & 0 & & \vdots & E_1^{(s)}-E_1^{(s-1)} \\[2pt] E_2^{(s-1)} & E_1^{(s-1)} & 1 & \ddots & \vdots & E_2^{(s)}-E_2^{(s-1)} \\[2pt] \vdots & E_2^{(s-1)} & \ddots & \ddots & 0 & \vdots \\[2pt] \vdots & \vdots & \ddots & E_1^{(r+1)} & 1 & \vdots \\[2pt] E_{s-r+1}^{(s-1)} & E_{s-r}^{(s-1)} & \cdots & E_2^{(r+1)} & E_1^{(r)} & E_{s-r+1}^{(s)}-E_{s-r+1}^{(s-1)} \end{array} \right| \\[5pt] & \quad \qquad \text{(by changing the first column and the last column)} \\&= -\left| \begin{array}{@{\,}cccccc@{\,}} 1 & 0 & \cdots & \cdots & 0 & E_{1}^{(s)}-E_{1}^{(s-1)} \\ E_1^{(s-1)} & 1 & 0 & & \vdots & E_{2}^{(s)}-E_{2}^{(s-1)} \\ E_2^{(s-1)} & E_1^{(s-2)} & 1 & \ddots & \vdots & E_{3}^{(s)}-E_{3}^{(s-1)} \\ \vdots & \ddots & \ddots & \ddots & 0 & \vdots \\ \vdots & & \ddots & E_1^{(r+1)} & 1 & E_{s-r}^{(s)}-E_{s-r}^{(s-1)} \\ E_{s-r}^{(s-1)} & \cdots & \cdots & E_2^{(r+1)} & E_1^{(r)} & E_{s-r+1}^{(s)}-E_{s-r+1}^{(s-1)} \end{array} \right| \end{align*} $$

for $1 \leq r < s < n$ , which is $-q_{rs}$ from Lemma 5.1. This completes the proof.

It follows from Proposition 5.2 that the image of $\{-F_{i,j} \mid j \in [n-1] \ \text {and} \ h(j) < i \leq n \}$ under the map $\varphi $ in equation (5.2) is $\{q_{rs} \mid 2 \leq s \leq n \ \text {and} \ 1 \leq r \leq n-h(n+1-s) \}$ . Thus, the surjective map $\varphi $ induces the surjective homomorphism

(5.5) $$ \begin{align} &\varphi_h: {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text{and} \ h(j) < i \leq n) \\ & \qquad \twoheadrightarrow Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text{and} \ 1 \leq r \leq n-h(n+1-s)); \ x_{ij} \mapsto \, ^h E_{i-j}^{(n-j)}. \notag \end{align} $$

6 Hilbert series

In order to prove that $\varphi _h$ in equation (5.5) is an isomorphism, we introduce certain degrees for the variables $\{x_{ij} \mid 1 \leq j < i \leq n \}$ and $\{x_1,\ldots ,x_n, q_{rs} \mid 1 \leq r < s \leq n \}$ so that the two sides of equation (5.5) are graded ${\mathbb {C}}$ -algebras. We then show that the two sides have identical Hilbert series.

Let ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]$ and ${\mathbb {C}}[x_1,\ldots ,x_n,q_{rs} \mid 1 \leq r < s \leq n]$ be the polynomial rings equipped with a grading defined by

(6.1) $$ \begin{align} \!\!\!\deg x_{ij} &= 2(i-j) \ \ \ \text{for} \ 1 \leq j < i \leq n;\quad \end{align} $$

(6.2) $$ \begin{align} \!\!\deg x_s &= 2 \ \ \ \text{for} \ s \in [n];\qquad\qquad\ \qquad \end{align} $$

(6.3) $$ \begin{align} \deg q_{rs} &= 2(s-r+1) \ \ \ \text{for} \ 1 \leq r < s \leq n. \end{align} $$

Proof. We prove this by induction on s. The base case is $s=1$ , which is clear since $E_1^{(1)}=x_1$ . Now, suppose that $s> 1$ and assume by induction that the claim is true for arbitrary $s'$ with $s' \leq s-1$ . From Lemma 4.8, we have

$$ \begin{align*} E_r^{(s)} = E_r^{(s-1)} + E_{r-1}^{(s-1)} x_s + \sum_{k=1}^{r-1} E_{r-1-k}^{(s-1-k)} q_{s-k \, s}. \end{align*} $$

By the inductive hypothesis with equations (6.2) and (6.3), one can see that $E_r^{(s)}$ is homogeneous of degree $2r$ .

In order to see that the polynomial $F_{i,j}$ is homogeneous, we introduce the following polynomials.

Definition 6.3. Let $1 \leq j \leq n-1$ and $j \leq m_j <n$ . For $i>m_j$ , we define polynomials $\tilde F^{\langle m_j\rangle }_{i,j}$ by

Here, we take the convention that $x_{n+1\,j}=0$ for $j \in [n-1]$ .

By equations (3.3) and (3.4) one has

(6.4) $$ \begin{align} F_{i,j} = \tilde F^{\langle i-1 \rangle}_{i,j} \end{align} $$

for $1 \leq j < i \leq n$ .

Proof. It suffices to show that $\tilde F^{\langle m_j\rangle }_{i,j}$ is homogeneous of degree $2(i-j+1)$ for $j \leq m_j <i$ by equation (6.4). At first, we fix j. We prove this statement by induction on $m_j$ with this fixed j. The base case is $m_j =j$ . For arbitrary $i> j$ , since we have

$$ \begin{align*} \tilde F^{\langle 1 \rangle}_{i,1}&=\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{i1} & x_{i+1 \, 1} \end{array} \right| = x_{i+1 \, 1} - x_{i1}x_{21} \ \text{for} \ j=1; \\ \tilde F^{\langle j \rangle}_{i,j}&=\left| \begin{array}{@{\,}ccc@{\,}} 1 & 0 & 1 \\ x_{j \, j-1} & 1 & x_{j+1 \,j} \\ x_{i \, j-1} & x_{ij} & x_{i+1 \, j} \end{array} \right| = x_{i+1 \, j} + x_{j \, j-1}x_{ij} - x_{i \, j-1} - x_{j+1 \, j}x_{ij} \ \text{for} \ j>1, \end{align*} $$

one can easily see from equation (6.1) that $\tilde F^{\langle j \rangle }_{i,j}$ is homogeneous of degree $2(i-j+1)$ . This shows the base case.

We proceed to the inductive step. Suppose now that $m_j> j$ and that the claim holds for $m_j-1$ with any allowable choices of the first subscript $i'$ in $\tilde F^{\langle m_j -1\rangle }_{i',j}$ . By the cofactor expansion along the second-to-last column, we have

$$ \begin{align*} \tilde F^{\langle m_j\rangle}_{i,j} = \tilde F^{\langle m_j-1\rangle}_{i,j} - x_{i \, m_j} \tilde F^{\langle m_j-1\rangle}_{m_j,j}. \end{align*} $$

By the inductive hypothesis and equation (6.1), the right-hand side above is homogeneous of degree $2(i-j+1)$ , as desired. This completes the proof.

Recall from the end of Section 5 that we constructed the map $\varphi _h$ in equation (5.5) from ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text {and} \ h(j) < i \leq n)$ to $Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text {and} \ 1 \leq r \leq n-h(n+1-s))$ . One can see from Lemmas 6.2 and 6.4 that these are graded ${\mathbb {C}}$ -algebras. (Note that $Q_n$ is also a graded ${\mathbb {C}}$ -algebra.) For the rest of this section, we prove that these graded ${\mathbb {C}}$ -algebras have the same Hilbert series.

Definition 6.5. Let $R=\oplus _{i=0}^\infty R_i$ be a graded ${\mathbb {C}}$ -algebra where each homogeneous component $R_i$ of degree i is a finite-dimensional vector space over ${\mathbb {C}}$ . Then its Hilbert series is defined to be

A sequence of homogeneous polynomials $\theta _1,\ldots ,\theta _p \in R$ of positive degrees is a regular sequence in R if $\theta _k$ is a nonzero divisor of $R/(\theta _1,\ldots ,\theta _{k-1})$ for all $1 \leq k \leq p$ .

The following facts are well known in commutative algebra. See [Reference Stanley35, Chapter I, Section 5]. (See also [Reference Fukukawa, Harada and Masuda18, Proposition 5.1].)

Lemma 6.6. Let $R=\oplus _{i=0}^\infty R_i$ be a graded ${\mathbb {C}}$ -algebra with $\dim _{\mathbb {C}} R_i < \infty $ for each i. A sequence of homogeneous polynomials $\theta _1,\ldots ,\theta _p \in R$ of positive degrees is a regular sequence in R if and only if the Hilbert series of $R/(\theta _1,\ldots ,\theta _p)$ is given by

$$ \begin{align*} \operatorname{\mathrm{Hilb}}(R/(\theta_1,\ldots,\theta_p),t) = \operatorname{\mathrm{Hilb}}(R,t) \cdot \prod_{k=1}^p (1-t^{\deg \theta_k}). \end{align*} $$

We remark that the number of the homogeneous polynomials $\theta _1,\ldots ,\theta _n$ is equal to the number of the variables $y_1,\ldots ,y_n$ in the polynomial ring ${\mathbb {C}}[y_1,\ldots ,y_n]$ in Lemma 6.7. By using two lemmas above, we compute the Hilbert series of ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text {and} \ h(j) < i \leq n)$ and $Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text {and} \ 1 \leq r \leq n-h(n+1-s))$ .

Proposition 6.9. The Hilbert series of ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text {and} \ h(j) < i \leq n)$ equipped with a grading in equation (6.1) is equal to

$$ \begin{align*} &\operatorname{\mathrm{Hilb}}({\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text{and} \ h(j) < i \leq n),t) \\ = &\prod_{1 \leq j \leq n-1 \atop j < i \leq h(j)} \frac{1}{1-t^{2(i-j+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+\cdots+t^{2k}). \end{align*} $$

Proof. Since a subsequence of a regular sequence is also a regular sequence from the definition of a regular sequence, by Lemma 6.8, the polynomials $F_{i,j} \ (j \in [n-1], h(j) < i \leq n)$ form a regular sequence in the polynomial ring ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]$ . Thus, it follows from Lemma 6.6 that

(6.5) $$ \begin{align} &\operatorname{\mathrm{Hilb}}({\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text{and} \ h(j) < i \leq n),t) \\[5pt] = &\operatorname{\mathrm{Hilb}}({\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n],t) \cdot \prod_{1 \leq j \leq n-1 \atop h(j) < i \leq n} (1-t^{\deg F_{i,j}}) \notag \\ = &\prod_{1 \leq j < i \leq n} \frac{1}{1-t^{2(i-j)}} \cdot \prod_{1 \leq j \leq n-1 \atop h(j) < i \leq n} (1-t^{2(i-j+1)}) \ \ \ \ \ \text{(by Lemma~}6.4). \notag \end{align} $$

Here, we note that

$$ \begin{align*} \prod_{1 \leq j < i \leq n} \frac{1}{1-t^{2(i-j)}} \cdot (1-t^2)^{n-1} = \prod_{1 \leq j < i \leq n} \frac{1}{1-t^{2(i-j+1)}} \cdot (1-t^4)(1-t^6) \cdots (1-t^{2n}). \end{align*} $$

In fact, exponents appeared on the left-hand side and exponents on the right-hand side are described as numbers in shaded boxes of the left figure and the right figure in Figure 4, respectively.

Figure 4 The values $2(i-j)$ and $2(i-j+1)$ for $1 \leq j < i \leq n$

This equality leads us to the equality

$$ \begin{align*} \prod_{1 \leq j < i \leq n} \frac{1}{1-t^{2(i-j)}} = \prod_{1 \leq j < i \leq n} \frac{1}{1-t^{2(i-j+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+ \cdots +t^{2k}), \end{align*} $$

so by equation (6.5), one has

$$ \begin{align*} &\operatorname{\mathrm{Hilb}}({\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(F_{i,j} \mid j \in [n-1] \ \text{and} \ h(j) < i \leq n),t) \\[5pt] = &\prod_{1 \leq j < i \leq n} \frac{1}{1-t^{2(i-j+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+ \cdots +t^{2k}) \cdot \prod_{1 \leq j \leq n-1 \atop h(j) < i \leq n} (1-t^{2(i-j+1)}) \\[5pt] = &\prod_{1 \leq j \leq n-1 \atop j < i \leq h(j)} \frac{1}{1-t^{2(i-j+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+\cdots+t^{2k}), \end{align*} $$

as desired.

Proposition 6.11. The Hilbert series of $Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text {and} \ 1 \leq r \leq n-h(n+1-s))$ equipped with a grading in equations (6.2) and (6.3) is equal to

$$ \begin{align*} &\operatorname{\mathrm{Hilb}}(Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text{and} \ 1 \leq r \leq n-h(n+1-s)),t) \\[5pt] = &\prod_{1 \leq j \leq n-1 \atop j < i \leq h(j)} \frac{1}{1-t^{2(i-j+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+\cdots+t^{2k}). \end{align*} $$

Proof. Recall from the definition (5.1) that

$$ \begin{align*} Q_n = {\mathbb{C}}[x_1,\ldots,x_n, q_{rs} \mid 1 \leq r < s \leq n]/(E_1^{(n)}, \ldots, E_n^{(n)}). \end{align*} $$

Lemma 6.10 implies that the monomials $q_{rs} \ (2 \leq s \leq n, 1 \leq r \leq n-h(n+1-s))$ form a regular sequence in $Q_n$ by the definition of a regular sequence. Hence, by Lemma 6.6, we have

(6.6) $$ \begin{align} &\operatorname{\mathrm{Hilb}}(Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text{and} \ 1 \leq r \leq n-h(n+1-s)),t) \\[5pt] = &\operatorname{\mathrm{Hilb}}(Q_n,t) \cdot \prod_{2 \leq s \leq n \atop 1 \leq r \leq n-h(n+1-s)} (1-t^{\deg q_{rs}}) \notag \\[5pt] = &\operatorname{\mathrm{Hilb}}(Q_n,t) \cdot \prod_{2 \leq s \leq n \atop 1 \leq r \leq n-h(n+1-s)} (1-t^{2(s-r+1)}). \notag \end{align} $$

Since a subsequence of a regular sequence is again a regular sequence from the definition of a regular sequence, the polynomials $E_1^{(n)},\ldots ,E_n^{(n)}$ form a regular sequence in the polynomial ring ${\mathbb {C}}[x_1,\ldots ,x_n, q_{rs} \mid 1 \leq r < s \leq n]$ by Lemma 6.10. By using Lemma 6.6 again, one has

(6.7) $$ \begin{align} \operatorname{\mathrm{Hilb}}(Q_n,t) =& \operatorname{\mathrm{Hilb}}({\mathbb{C}}[x_1,\ldots,x_n, q_{rs} \mid 1 \leq r < s \leq n],t) \cdot \prod_{k=1}^n (1-t^{\deg E_k^{(n)}}) \\[5pt] = &\frac{1}{(1-t^2)^n} \cdot \prod_{1 \leq r < s \leq n} \frac{1}{1-t^{2(s-r+1)}} \cdot \prod_{k=1}^n (1-t^{2k}) \ \ \ \ \ \text{(by Lemma~}6.2) \notag \\[5pt] =& \prod_{1 \leq r < s \leq n} \frac{1}{1-t^{2(s-r+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+ \cdots +t^{2k}). \notag \end{align} $$

By equations (6.6) and (6.7), we obtain

$$ \begin{align*} &\operatorname{\mathrm{Hilb}}(Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text{and} \ 1 \leq r \leq n-h(n+1-s)),t) \\ = & \prod_{1 \leq r < s \leq n} \frac{1}{1-t^{2(s-r+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+ \cdots +t^{2k}) \cdot \prod_{2 \leq s \leq n \atop 1 \leq r \leq n-h(n+1-s)} (1-t^{2(s-r+1)}) \\ = & \prod_{1 \leq j < i \leq n} \frac{1}{1-t^{2(i-j+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+ \cdots +t^{2k}) \cdot \prod_{1 \leq j \leq n-1 \atop h(j)+1 \leq i \leq n} (1-t^{2(i-j+1)}) \\ & \hspace{100pt} (\text{by setting} \ i=n+1-r \ \text{and} \ j=n+1-s \ \text{in the third product}) \\ = &\prod_{1 \leq j \leq n-1 \atop j < i \leq h(j)} \frac{1}{1-t^{2(i-j+1)}} \cdot \prod_{k=1}^{n-1} (1+t^2+t^4+\cdots+t^{2k}), \end{align*} $$

as desired.

7 Proof of Theorem 4.13

We now prove Theorem 4.13.

Proof of Theorem 4.13.

We first note that there exists a canonical isomorphism

$$ \begin{align*} &Q_n/(q_{rs} \mid 2 \leq s \leq n \ \text{and} \ 1 \leq r \leq n-h(n+1-s)) \\ \cong & {\mathbb{C}}[x_1,\ldots,x_n, q_{rs} \mid 2 \leq s \leq n, n-h(n+1-s)<r<s]/(^h E_1^{(n)}, \ldots, \, ^h E_n^{(n)}) \end{align*} $$

by Definition 4.10 and equation (5.1). Under the identification above and the presentation in equation (3.7), we can rewrite $\varphi _h$ in equation (5.5) as

$$ \begin{align*} \varphi_h: \Gamma(\mathcal{Z}(N,h)_e, \mathcal{O}_{\mathcal{Z}(N,h)_e}) \twoheadrightarrow \frac{{\mathbb{C}}[x_1,\ldots,x_n, q_{rs} \mid 2 \leq s \leq n, n-h(n+1-s)<r<s]}{(^h E_1^{(n)}, \ldots, \, ^h E_n^{(n)})}, \end{align*} $$

which is defined by

$$ \begin{align*} \varphi_h(x_{ij}) = \, ^h E_{i-j}^{(n-j)}. \end{align*} $$

Here, by slightly abuse of notation we used the same symbol $\varphi _h$ for the map above. The map $\varphi _h$ is surjective and this preserves the gradings on both graded ${\mathbb {C}}$ -algebras from equation (6.1) and Lemma 6.2. It follows from Propositions 6.9 and 6.11 that the two sides of $\varphi _h$ have identical Hilbert series. Therefore, we conclude that $\varphi _h$ is an isomorphism.

Corollary 7.2. There is an isomorphism of ${\mathbb {C}}$ -algebras

$$ \begin{align*} \Gamma(\mathcal{Z}(N,id)_e, \mathcal{O}_{\mathcal{Z}(N,id)_e}) \cong H^*(Fl({\mathbb{C}}^n)). \end{align*} $$

Proof. Applying the isomorphism (4.5) to the case when $h=id$ , we obtain

$$ \begin{align*} \Gamma(\mathcal{Z}(N,id)_e, \mathcal{O}_{\mathcal{Z}(N,id)_e}) \cong {\mathbb{C}}[x_1,\ldots,x_n]/(e_1(x_1,\ldots,x_n), \ldots, e_n(x_1,\ldots,x_n)), \end{align*} $$

where $e_i(x_1, \ldots , x_n)$ denotes the i-th elementary symmetric polynomial in the variables $x_1,\ldots ,x_n$ . As is well known, the right hand side above is a presentation for the cohomology ring $H^*(Fl({\mathbb {C}}^n))$ (e.g., [Reference Fulton19, Section 10.2, Proposition 3]).

We constructed the isomorphism of graded ${\mathbb {C}}$ -algebras

(7.1) $$ \begin{align} \varphi_h: \Gamma(\mathcal{Z}(N,h)_e, \mathcal{O}_{\mathcal{Z}(N,h)_e}) \xrightarrow{\cong} \frac{{\mathbb{C}}[x_1,\ldots,x_n, q_{rs} \mid 2 \leq s \leq n, n-h(n+1-s)<r<s]}{(^h E_1^{(n)}, \ldots, \, ^h E_n^{(n)})}, \end{align} $$

which sends $x_{ij}$ to $^h E_{i-j}^{(n-j)}$ for all $1 \leq j < i \leq n$ from the homomorphism $\varphi $ in equation (5.2). The following result follows from Proposition 5.2.

Proposition 7.3. The inverse map of $\varphi _h$ in equation (7.1) is given by

$$ \begin{align*} \varphi_h^{-1}(x_s) &= x_{n-s+1 \, n-s} - x_{n-s+2 \, n-s+1} \ \ \ \text{for} \ s \in [n] \\ \varphi_h^{-1}(q_{rs}) &= -F_{n+1-r,n+1-s} \ \ \ \text{for} \ 2 \leq s \leq n \ \text{and} \ n-h(n+1-s)<r<s \end{align*} $$

with the convention that $x_{n-s+2 \, n-s+1}=0$ whenever $s=1$ and $x_{n-s+1 \, n-s}=0$ whenever $s=n$ .

We conclude the following result from the discussion above.

Corollary 7.4. Let $h:[n] \rightarrow [n]$ be a Hessenberg function. Then the following commutative diagram holds

where both vertical arrows denote the canonical surjective maps under the presentation (3.7). In particular, if h is indecomposable, then we obtain the following commutative diagram

where the left vertical arrow is induced from the inclusion $\operatorname {\mathrm {Hess}}(N,h) \cap \Omega _e^\circ \hookrightarrow Fl({\mathbb {C}}^n) \cap \Omega _e^\circ $ . Note that both vertical arrows are surjective.

8 Jacobian matrix

It is an interesting and challenging problem to find an explicit description of the singular locus of (regular nilpotent) Hessenberg varieties $\operatorname {\mathrm {Hess}}(N,h)$ . There are already partial results for the problem stated above in [Reference Abe and Insko3, Reference Escobar, Precup and Shareshian15, Reference Insko and Yong26]. For the rest of the paper, we will analyze the singular locus of $\operatorname {\mathrm {Hess}}(N,h) \cap \Omega _e^\circ $ for some Hessenberg function h as an application of our result. The isomorphism (4.6) in Theorem 4.13 yields that if h is indecomposable, then the singular locus of the open set $\operatorname {\mathrm {Hess}}(N,h) \cap \Omega _e^\circ $ in $\operatorname {\mathrm {Hess}}(N,h)$ is isomorphic to the singular locus of the zero set defined by n polynomials $^h E_1^{(n)}, \ldots , \, ^h E_n^{(n)}$ in ${\mathbb {C}}^{n+\sum _{j=1}^n (h(j)-j))}$ with the variables $x_1,\ldots ,x_n, q_{rs} \ (2 \leq s \leq n, n-h(n+1-s)<r<s)$ . In this section, we give an explicit formula for partial derivatives $\partial E_i^{(n)}/\partial x_s \ (1 \leq s \leq n)$ and $\partial E_i^{(n)}/\partial q_{rs} \ (1 \leq r < s \leq n)$ for each $i \in [n]$ .

For positive integers a and b with $1 \leq a \leq b \leq n$ , we set

As in Definition 4.6, we introduce the following polynomials.

Definition 8.1. Let $M_{[a,b]}$ be the matrix of size $(b-a+1) \times (b-a+1)$ defined as

We define polynomials $E_1^{[a,b]}, \ldots , E_{b-a+1}^{[a,b]} \in {\mathbb {C}}[x_a,\ldots ,x_b,q_{rs} \mid a \leq r < s \leq b]$ by

$$ \begin{align*} \det(\lambda I_{b-a+1} - M_{[a,b]}) = \lambda^{b-a+1} - E_1^{[a,b]} \lambda^{b-a} + E_2^{[a,b]} \lambda^{b-a-1} + \cdots + (-1)^{b-a+1} E_{b-a+1}^{[a,b]}. \end{align*} $$

Note that if $a=1$ , then we obtain

(8.1) $$ \begin{align} E_i^{[1,b]} = E_i^{(b)} \ \ \ \text{for each} \ i \in [b] \end{align} $$

by the definition. In what follows, we use the symbol

and we take the following convention

(8.2)

for $b \geq a-1$ . By the same argument for the proof of Lemma 4.8, one can prove the lemma below. We leave the detail to the reader.

Lemma 8.2. Let $1 \leq a \leq b \leq n$ . For $1 \leq i \leq b-a+1$ , we have

$$ \begin{align*} E_i^{[a,b]} = E_i^{[a,b-1]} + E_{i-1}^{[a,b-1]} x_b + \sum_{k=1}^{i-1} E_{i-1-k}^{[a,b-1-k]} q_{b-k \, b} \end{align*} $$

with the convention that $\sum _{k=1}^{i-1} E_{i-1-k}^{[a,b-1-k]} q_{b-k \, b}=0$ whenever $i=1$ .

Proof. (1) By Definition 4.6, one has

(8.3) $$ \begin{align} \det(\lambda I_n - M_n) = \lambda^n - E_1^{(n)} \lambda^{n-1} + E_2^{(n)} \lambda^{n-2} + \cdots + (-1)^n E_n^{(n)}. \end{align} $$

We think of $\lambda $ as a variable in the equality above, and we partial differentiate the both sides with respect to $x_s$ . Then $\frac {\partial }{\partial x_s} E_i^{(n)}$ is equal to the coefficient of $\lambda ^{n-i}$ for $\frac {\partial }{\partial x_s}\det (\lambda I_n - M_n)$ multiplied by $(-1)^{i}$ . Since the variable $x_s$ appears in only the $(s,s)$ -th entry of the matrix $(\lambda I_n - M_n)$ , from the cofactor expansion along the s-th column for $\det (\lambda I_n - M_n)$ , we can write

$$ \begin{align*} \det(\lambda I_n - M_n) = (\lambda-x_s) \det(\lambda I_{s-1} - M_{[1,s-1]}) \det(\lambda I_{n-s} - M_{[s+1,n]}) + F \end{align*} $$

for some polynomial $F \in {\mathbb {C}}[x_1,\ldots ,\widehat {x_s},\ldots ,x_n,q_{ij} \mid 1 \leq i < j \leq n]$ . Here, the caret sign $\widehat {}$ over $x_s$ means that the entry is omitted. Hence, we obtain

$$ \begin{align*} \frac{\partial}{\partial x_s}\det(\lambda I_n - M_n) &= - \det(\lambda I_{s-1} - M_{[1,s-1]}) \det(\lambda I_{n-s} - M_{[s+1,n]}) \\ &= - \left(\sum_{u=0}^{s-1} (-1)^u E_u^{[1,s-1]} \lambda^{s-1-u} \right) \left( \sum_{v=0}^{n-s} (-1)^v E_v^{[s+1,n]} \lambda^{n-s-v} \right) \\ & \hspace{30pt} \text{(by Definition~}8.1) \\ &= - \sum_{u=0}^{s-1} \sum_{v=0}^{n-s} (-1)^{u+v} E_u^{[1,s-1]}E_v^{[s+1,n]} \lambda^{n-1-(u+v)} \\ &= - \sum_{\ell=0}^{n-1} \left( \sum_{k=0}^{\ell} (-1)^{\ell} E_{\ell-k}^{[1,s-1]}E_k^{[s+1,n]} \right) \lambda^{n-1-\ell}. \end{align*} $$

Therefore, we conclude that the coefficient of $\lambda ^{n-i}$ for $\frac {\partial }{\partial x_s}\det (\lambda I_n - M_n)$ multiplied by $(-1)^{i}$ is equal to

$$ \begin{align*} \frac{\partial}{\partial x_s} E_i^{(n)} = \sum_{k=0}^{i-1} E_{i-1-k}^{[1,s-1]}E_k^{[s+1,n]}, \end{align*} $$

as desired.

(2) We partial differentiate the both sides of equation (8.3) with respect to $q_{rs}$ , then $\frac {\partial }{\partial q_{rs}} E_j^{(n)}$ is equal to the coefficient of $\lambda ^{n-j}$ for $\frac {\partial }{\partial q_{rs}}\det (\lambda I_n - M_n)$ multiplied by $(-1)^{j}$ . Since the variable $q_{rs}$ appears in only the $(r,s)$ -th entry of the matrix $(\lambda I_n - M_n)$ , one can see from similar arguments as in the previous case that

$$ \begin{align*} \frac{\partial}{\partial q_{rs}}\det(\lambda I_n - M_n) &= (-1)^{r+s} \frac{\partial}{\partial q_{rs}} (-q_{rs}) \det(\lambda I_{r-1} - M_{[1,r-1]}) \det(\lambda I_{n-s} - M_{[s+1,n]}) \\ &= (-1)^{r+s+1} \left(\sum_{u=0}^{r-1} (-1)^u E_u^{[1,r-1]} \lambda^{r-1-u} \right) \left( \sum_{v=0}^{n-s} (-1)^v E_v^{[s+1,n]} \lambda^{n-s-v} \right) \\ &= (-1)^{r+s+1} \sum_{u=0}^{r-1} \sum_{v=0}^{n-s} (-1)^{u+v} E_u^{[1,r-1]}E_v^{[s+1,n]} \lambda^{n-1-(u+v+s-r)} \\ &= (-1)^{r+s+1} \sum_{\ell=s-r}^{n-1} \left( \sum_{k=0}^{\ell-s+r} (-1)^{\ell-s+r} E_{\ell-s+r-k}^{[1,r-1]}E_k^{[s+1,n]} \right) \lambda^{n-1-\ell}. \end{align*} $$

Thus, the coefficient of $\lambda ^{n-j}$ for $\frac {\partial }{\partial q_{rs}}\det (\lambda I_n - M_n)$ multiplied by $(-1)^{j}$ is

$$ \begin{align*} \frac{\partial}{\partial q_{rs}} E_j^{(n)} = \begin{cases} 0 \ \ \ & \text{if} \ 1 \leq j \leq s-r, \\ \sum_{k=0}^{j-1-s+r} E_{j-1-s+r-k}^{[1,r-1]}E_k^{[s+1,n]} \ \ \ & \text{if} \ s-r+1 \leq j \leq n, \end{cases} \end{align*} $$

as desired. This completes the proof.

In what follows, we set

(8.4)

and we see the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} E_{i}^{(n)} \big )_{i,(r,s)}$ below.

Example 8.4. Let $n=3$ . Then the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} E_{i}^{(3)} \big )_{i,(r,s)}$ is described as

by Lemma 8.3. Also, the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} E_{i}^{(4)} \big )_{i,(r,s)}$ for $n=4$ is given by

The Jacobian matrices above have a full rank. In general, one can verify from Lemma 8.3 that the rank of the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} E_{i}^{(n)} \big )_{i,(r,s)}$ is full for arbitrary n. This fact also follows from the well-known fact that $Fl({\mathbb {C}}^n) \cap \Omega _e^\circ $ is smooth and Theorem 4.13 for the case when $h=(n,\ldots ,n)$ .

Let $h: [n] \to [n]$ be a Hessenberg function. As in Definition 4.10, for $0 \leq i \leq n$ and $a,b$ with $b \geq a-1$ , we define

(8.5)

Note that we take the convention in equation (8.2). By the definition (8.5), it is straightforward to see that for arbitrary Hessenberg function $h: [n] \to [n]$ , $i \in [n]$ , and $(r,s)$ with $2 \leq s \leq n, n-h(n+1-s)<r \leq s$ ,

$$ \begin{align*} \left. \frac{\partial}{\partial q_{rs}} \, ^h E_{i}^{(n)} = \left( \frac{\partial}{\partial q_{rs}} E_{i}^{(n)} \right) \right|_{q_{uv}=0 \ (2 \leq v \leq n \ \text{and} \ 1 \leq u \leq n-h(n+1-v))}. \end{align*} $$

Combining this and Lemma 8.3, we have that for $1 \leq r \leq s \leq n$ ,

(8.6) $$ \begin{align} \frac{\partial}{\partial q_{rs}} \, ^h E_{i+s-r}^{(n)} = \begin{cases} 0 & \text{if} \ 1-(s-r) \leq i \leq 0, \\ 1 & \text{if} \ i = 1, \\ \sum_{k=0}^{i-1} \, ^h E_{i-1-k}^{[1,r-1]} \, ^h E_{k}^{[s+1,n]} & \text{if} \ 2 \leq i \leq n-(s-r). \end{cases} \end{align} $$

For $2 \leq m \leq n-1$ , we define a Hessenberg function $h_m:[n] \rightarrow [n]$ by

(8.7) $$ \begin{align} h_m=(m,n,\ldots,n). \end{align} $$

The zero set of $\{^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\}$ in ${\mathbb {C}}^{\frac {1}{2}n(n+1)-(n-m)}$ with the variables $x_1.\ldots ,x_n,q_{rs} \ (2 \leq s \leq n, n-h_m(n+1-s)<r < s)$ is denoted by $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ , that is,

(8.8)

where .

Proposition 8.5. Let $2 \leq m \leq n-1$ and $h_m$ be the Hessenberg function in equation (8.7). Then, the singular locus of $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ in equation (8.8) is given by the solution set of the equations

$$ \begin{align*} \frac{\partial}{\partial q_{rs}} \, ^{h_m} E_{n}^{(n)} = 0 \ \ \ \text{for all} \ 2 \leq s \leq n \ \text{and} \ n-h_m(n+1-s)<r \leq s. \end{align*} $$

Here, we recall our convention (8.4) that $q_{rs}=x_s$ whenever $r=s$ .

Before we prove Proposition 8.5, we give an example of the singular locus of the zero set $V\big (\, ^{h_m} E_1^{(3)}, \, ^{h_m} E_2^{(3)}, \, ^{h_m} E_3^{(3)}\big )$ for $m=2$ by using Proposition 8.5.

Example 8.6. Consider the case when $h=(2,3,3)$ for $n=3$ . The Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} \, ^h E_{i}^{(3)} \big )_{i,(r,s) \neq (1,3)}$ is obtained from $\big (\frac {\partial }{\partial q_{rs}} E_{i}^{(3)} \big )_{i,(r,s)}$ by forgetting the quantum parameter $q_{13}$ . As seen in Example 8.4, the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} \, ^h E_{i}^{(3)} \big )_{i,(r,s) \neq (1,3)}$ is shown as

It follows from Proposition 8.5 that the singular locus of $V\big (\, ^h E_1^{(3)}, \, ^h E_2^{(3)}, \, ^h E_3^{(3)}\big )$ is given by the solution set of the equations $^h E_2^{[2,3]}=\, ^h E_1^{[1,1]} \, ^h E_1^{[3,3]}=\, ^h E_2^{[1,2]}=\, ^h E_1^{[3,3]}=\, ^h E_1^{[1,1]}=0$ . The resulting solution is as follows:

$$ \begin{align*} x_1=0, \ x_3=0, \ q_{12}=0, \ q_{23}=0. \end{align*} $$

Then it follows from equation (4.6) and Proposition 7.3 that the singular locus of $\operatorname {\mathrm {Pet}}_3 \cap \Omega _e^\circ $ is given by the solution set of the equations $x_{32}=0, \ x_{21}=0, \ F_{3,2}=0, \ F_{2,1}=0$ . This is equivalent to

$$ \begin{align*} x_{21}=0, \ x_{32}=0, \ x_{31}=0. \end{align*} $$

Hence, the singular locus of $\operatorname {\mathrm {Pet}}_3 \cap \Omega _e^\circ $ is $\{eB \}$ . Note that the singular locus of the Peterson variety $\operatorname {\mathrm {Pet}}_n$ is given by [Reference Insko and Yong26]. We will explain this in Appendix A.

Proof of Proposition 8.5.

Recall that a point in ${\mathbb {C}}^{\frac {1}{2}n(n+1)-(n-m)}$ is a singular point of $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ if and only if the rank of the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} \, ^{h_m} E_{i}^{(n)}(a,p) \big )_{i,(r,s)}$ is not full. It is clear that the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} \, ^{h_m} E_{i}^{(n)}(a,p) \big )_{i,(r,s)}$ is not full if the n-th row vector is zero. Thus, it is enough to prove that if a point $(a,p)$ in ${\mathbb {C}}^{\frac {1}{2}n(n+1)-(n-m)}$ is a singular point of $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ , then the n-th row vector $\big ( \frac {\partial }{\partial q_{rs}} \, ^{h_m} E_{n}^{(n)}(a,p) \big )_{(r,s)}$ of the Jacobian matrix is zero.

Since the column vector with respect to $\frac {\partial }{\partial q_{rs}}$ of the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} \, ^{h_m} E_{i}^{(n)}(a,p) \big )_{i,(r,s)}$ is of the form $(\underbrace {0,\ldots ,0}_{s-r},1,*,\ldots ,*)^t$ by equation (8.6), the first $n-1$ row vectors of the Jacobian matrix $\big (\frac {\partial }{\partial q_{rs}} \, ^{h_m} E_{i}^{(n)}(a,p) \big )_{i,(r,s)}$ are linearly independent. By the assumption that $\big (\frac {\partial }{\partial q_{rs}} \, ^{h_m} E_{i}^{(n)}(a,p) \big )_{i,(r,s)}$ does not have full rank, the n-th row vector must be written as a linear combination of the first $n-1$ row vectors, that is,

(8.9) $$ \begin{align} \left(\frac{\partial}{\partial q_{rs}} \, ^{h_m} E_{n}^{(n)}(a,p) \right)_{(r,s)} = \sum_{i=1}^{n-1} c_i \left(\frac{\partial}{\partial q_{rs}} \, ^{h_m} E_{i}^{(n)}(a,p) \right)_{(r,s)} \end{align} $$

for some $c_1,\ldots ,c_{n-1} \in {\mathbb {C}}$ . We note that the pair $(r,s)$ can be taken as $2 \leq s \leq n$ and $n-h_m(n+1-s)<r \leq s$ in the equality above. Recall that we denote the singular point $(a_1,\ldots ,a_n,p_{rs})_{2 \leq s \leq n, n-h_m(n+1-s)<r < s}$ of $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ by $(a,p)$ .

Claim 1. The coefficients $c_i$ of equation (8.9) must be $(-1)^{n-i+1} a_n^{n-i}$ for $i \in [n-1]$ .

We prove Claim 1 by descending induction on i. The base case is $i=n-1$ . Comparing the $(r,s)=(1,n-1)$ -th component of equation (8.9), we have

$$ \begin{align*} ^{h_m} E_1^{[n,n]}(a,p) = c_1 \cdot 0 + c_2 \cdot 0 + \cdots + c_{n-2} \cdot 0 + c_{n-1} \cdot 1 \end{align*} $$

by equation (8.6). Since $^{h_m} E_1^{[n,n]}=x_n$ , the equality above implies that $a_n=c_{n-1}$ . This shows the base case.

We now assume that $i<n-1$ and the assertion of the claim holds for arbitrary k with $k \geq i+1$ . It follows from equation (8.6) that the $(r,s)=(n-i,n-1)$ -th component of equation (8.9) gives

$$ \begin{align*} &^{h_m} E_{n-i-1}^{[1,n-i-1]}(a,p) \cdot \,^{h_m} E_1^{[n,n]}(a,p) \\=&c_i \cdot 1 + c_{i+1} \cdot \left( ^{h_m} E_1^{[1,n-i-1]}(a,p)+\,^{h_m} E_1^{[n,n]}(a,p) \right) \\&+ c_{i+2} \cdot \left( ^{h_m} E_2^{[1,n-i-1]}(a,p)+\,^{h_m} E_1^{[1,n-i-1]}(a,p)\cdot\,^{h_m} E_1^{[n,n]}(a,p) \right) \\&+ c_{i+3} \cdot \left( ^{h_m} E_3^{[1,n-i-1]}(a,p)+\,^{h_m} E_2^{[1,n-i-1]}(a,p)\cdot\,^{h_m} E_1^{[n,n]}(a,p) \right) \\&+ \cdots + c_{n-1} \cdot \left( ^{h_m} E_{n-i-1}^{[1,n-i-1]}(a,p)+\,^{h_m} E_{n-i-2}^{[1,n-i-1]}(a,p)\cdot\,^{h_m} E_1^{[n,n]}(a,p) \right). \end{align*} $$

Since $^{h_m} E_1^{[n,n]}(a,p)=a_n$ and $c_k=(-1)^{n-k+1} a_n^{n-k}$ for all $i+1 \leq k \leq n-1$ by our inductive assumption, $^{h_m} E_{n-i-1}^{[1,n-i-1]}(a,p) \cdot a_n$ is equal to

$$ \begin{align*} &c_i + \sum_{k=1}^{n-i-1} (-1)^{n-i-k+1} a_n^{n-i-k} \left( ^{h_m} E_k^{[1,n-i-1]}(a,p)+\,^{h_m} E_{k-1}^{[1,n-i-1]}(a,p)\cdot a_n \right) \\= & c_i + (-1)^{n-i} a_n^{n-i-1} \cdot a_n + a_n \cdot \,^{h_m} E_{n-i-1}^{[1,n-i-1]}(a,p). \end{align*} $$

This yields that $c_i=(-1)^{n-i+1} a_n^{n-i}$ and we proved Claim 1.

It follows from Claim 1 and equation (8.9) that

(8.10) $$ \begin{align} \left(\frac{\partial}{\partial q_{rs}} \, ^{h_m} E_{n}^{(n)}(a,p) \right)_{(r,s)} = \sum_{i=1}^{n-1} (-1)^{n-i+1} a_n^{n-i} \left(\frac{\partial}{\partial q_{rs}} \, ^{h_m} E_{i}^{(n)}(a,p) \right)_{(r,s)} \end{align} $$

for all $2 \leq s \leq n$ and $n-h_m(n+1-s)<r \leq s$ .

Claim 2. It holds that $p_{in}=0$ for all $n-m+1 \leq i \leq n-1$ .

We show Claim 2 by descending induction on i. The base case is $i=n-1$ . The $(r,s)=(1,n-2)$ -th component of equation (8.10) is equal to

$$ \begin{align*} ^{h_m} E_2^{[n-1,n]}(a,p) = -a_n^2 \cdot 1 + a_n \cdot \, ^{h_m} E_1^{[n-1,n]}(a,p) \end{align*} $$

from equation (8.6). Since $^{h_m} E_2^{[n-1,n]}(a,p)=a_{n-1}a_n+p_{n-1 \, n}$ and $^{h_m} E_1^{[n-1,n]}(a,p)=a_{n-1}+a_n$ , we have $p_{n-1 \, n}=0$ , which proves the base case.

Suppose now that $i < n-1$ and that the claim holds for any k with $k \geq i+1$ , that is, $p_{kn}=0$ for all $i+1 \leq k \leq n-1$ . By equation (8.6), the $(r,s)=(1,i-1)$ -th component of equation (8.10) is

(8.11) $$ \begin{align} ^{h_m} E_{n-i+1}^{[i,n]}(a,p) = &(-1)^{n-i+2}a_n^{n-i+1} \cdot 1 + (-1)^{n-i+1}a_n^{n-i} \cdot \, ^{h_m} E_1^{[i,n]}(a,p) \\ &+ (-1)^{n-i}a_n^{n-i-1} \cdot \, ^{h_m} E_2^{[i,n]}(a,p) +\cdots+a_n \cdot \, ^{h_m} E_{n-i}^{[i,n]}(a,p). \notag \end{align} $$

Here, we note that

$$ \begin{align*} \det (\lambda I_{n-i+1} - M_{[i,n]})|_{q_{kn}=0 \ (i+1 \leq k \leq n-1)} = (-1)^{n-i} (-q_{in}) + \det (\lambda I_{n-i} - M_{[i,n-1]}) \cdot (\lambda-x_n) \end{align*} $$

by the cofactor expansion along the last column. The left-hand side is written as

$$ \begin{align*} \sum_{\ell=0}^{n-i+1} \big( (-1)^\ell E_\ell^{[i,n]}|_{q_{kn}=0 \ (i+1 \leq k \leq n-1)} \big) \lambda^{n-i+1-\ell}, \end{align*} $$

and the right-hand side is

$$ \begin{align*} \lambda^{n-i+1} + \sum_{\ell=1}^{n-i} \big( (-1)^\ell (x_n E_{\ell-1}^{[i,n-1]} + E_\ell^{[i,n-1]}) \big) \lambda^{n-i+1-\ell} + (-1)^{n-i+1} (x_n E_{n-i}^{[i,n-1]} + q_{in}) \end{align*} $$

by definition. Thus, we obtain that

$$ \begin{align*} E_\ell^{[i,n]}|_{q_{kn}=0 \ (i+1 \leq k \leq n-1)} &= x_n E_{\ell-1}^{[i,n-1]} + E_\ell^{[i,n-1]} \ \ \ \text{for} \ 1 \leq \ell \leq n-i; \\ E_{n-i+1}^{[i,n]}|_{q_{kn}=0 \ (i+1 \leq k \leq n-1)} &=x_n E_{n-i}^{[i,n-1]} + q_{in}. \end{align*} $$

In particular, by our inductive hypothesis $p_{kn}=0$ for all $i+1 \leq k \leq n-1$ , one has

$$ \begin{align*} ^{h_m} E_\ell^{[i,n]}(a,p) &= a_n \, ^{h_m} E_{\ell-1}^{[i,n-1]}(a,p) + \, ^{h_m} E_\ell^{[i,n-1]}(a,p) \ \ \ \text{for} \ 1 \leq \ell \leq n-i; \\ ^{h_m} E_{n-i+1}^{[i,n]}(a,p) &=a_n \, ^{h_m} E_{n-i}^{[i,n-1]}(a,p) + p_{in}. \end{align*} $$

Substituting these equalities to equation (8.11), the left-hand side of equation (8.11) is

$$ \begin{align*} a_n \, ^{h_m} E_{n-i}^{[i,n-1]}(a,p) + p_{in}. \end{align*} $$

On the other hand, the right-hand side of equation (8.11) is

$$ \begin{align*} &(-1)^{n-i+2}a_n^{n-i+1} + \sum_{k=1}^{n-i} (-1)^{n-i-k+2} a_n^{n-i-k+1} \cdot \left(a_n \, ^{h_m} E_{k-1}^{[i,n-1]}(a,p)+\,^{h_m} E_{k}^{[i,n-1]}(a,p) \right) \\ = & a_n \cdot \, ^{h_m} E_{n-i}^{[i,n-1]}(a,p). \end{align*} $$

Hence, we obtain that $p_{in}=0$ as desired. This proves Claim 2.

Claim 3. We have $a_n=0$ .

It follows from Theorem 4.13 that

$$ \begin{align*} \operatorname{\mathrm{Hess}}(N,h_m) \cap \Omega_e^\circ \cong V\big(\, ^{h_m} E_1^{(n)}, \ldots, \, ^{h_m} E_n^{(n)}\big). \end{align*} $$

We denote by $b=(b_{ij})_{1 \leq j < i \leq n} \in \operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ \subset {\mathbb {C}}^{\frac {1}{2}n(n-1)}$ the image of the singular point $(a,p)$ of $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ under the isomorphism above. One can see from Proposition 7.3 and Claim 2 that

(8.12) $$ \begin{align} F_{i,1}(b)=-q_{n+1-i \, n}(a,p)=-p_{n+1-i \, n}=0 \ \ \ \text{for} \ 2 \leq i \leq m. \end{align} $$

Since $b \in \operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ $ , we also have

(8.13) $$ \begin{align} F_{i,1}(b)=0 \ \ \ \text{for} \ m+1 \leq i \leq n \end{align} $$

by Lemma 2.3. It follows from equations (8.12) and (8.13) and Lemma 2.3 again that the point ${b=(b_{ij})_{1 \leq j < i \leq n}}$ belongs to $\operatorname {\mathrm {Hess}}(N,h_1) \cap \Omega _e^\circ $ , where $h_1$ is the decomposable Hessenberg function defined by . As seen in Definition 3.2 and surrounding discussion, every flag $V_\bullet \in \operatorname {\mathrm {Hess}}(N,h_1)$ has $V_1 = {\mathbb {C}} \cdot (1,0,\ldots ,0)^t$ which implies that $b_{i1}=0$ for all $2 \leq i \leq n$ . It then follows from Proposition 7.3 that

$$ \begin{align*} a_n=x_n(a,p)=-x_{21}(b)=-b_{21}=0, \end{align*} $$

as desired. We have proven Claim 3.

Combining Claim 3 and equation (8.10), we conclude that the n-th row vector $\big ( \frac {\partial }{\partial q_{rs}} \, ^{h_m} E_{n}^{(n)}(a,p) \big )_{(r,s)}$ of the Jacobian matrix is zero. This completes the proof.

By using Proposition 8.5, we will explicitly describe the singular locus of $\operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ $ in Section 10. For this purpose, we will first study the singularity of $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ in next section.

9 Cyclic quotient singularity

In this section, we analyze the singularity of $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ , where $h_2=(2,n,\ldots ,n)$ . In fact, we see that the singularity is related with a cyclic quotient singularity. Then we can explicitly describe the singular locus of $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ .

First, we study relations between $F_{i,j}$ ’s and $\tilde F^{\langle m_j\rangle }_{i,j}$ ’s, which are defined in equations (3.3) and (3.4), and Definition 6.3, respectively.

Lemma 9.1. Let $1 \leq j \leq n-1$ and $j \leq m_j <n$ . For $i>m_j$ , we have

$$ \begin{align*} F_{i,j} = \tilde F^{\langle m_j\rangle}_{i,j} - \sum_{\ell=m_j+1}^{i-1} x_{i\ell} F_{\ell,j}, \end{align*} $$

where we take the convention that $\sum _{\ell =m_j+1}^{m_j} x_{i\ell } F_{\ell ,j}=0$ .

Proof. It suffices to show that

(9.1) $$ \begin{align} \tilde F^{\langle i-k \rangle}_{i,j} = \tilde F^{\langle m_j \rangle}_{i,j} - \sum_{\ell=m_j+1}^{i-k} x_{i\ell} F_{\ell,j} \ \ \ \text{for} \ 1 \leq k \leq i-m_j. \end{align} $$

Indeed, equation (9.1) for $k=1$ is the desired equality by equation (6.4). We prove equation (9.1) by descending induction on k. The base case is $k=i-m_j$ , which is clear. Now, suppose that $k < i-m_j$ and assume by induction that the claim is true for $k+1$ , that is,

(9.2) $$ \begin{align} \tilde F^{\langle i-k-1 \rangle}_{i,j} = \tilde F^{\langle m_j \rangle}_{i,j} - \sum_{\ell=m_j+1}^{i-k-1} x_{i\ell} F_{\ell,j}. \end{align} $$

Then we show equation (9.1). The left-hand side of equation (9.1) is equal to

$$ \begin{align*} \tilde F^{\langle i-k-1 \rangle}_{i,j} - x_{i\,i-k} F_{i-k,j} \end{align*} $$

by using the cofactor expansion along the second-to-last column. Combining this with the inductive hypothesis (9.2), we have proven equation (9.1).

Proposition 9.2. Let $1 \leq j \leq n-1$ and $j \leq m_j<n$ . For $i>m_j$ , the ideal

$$ \begin{align*} (F_{m_j+1,j},F_{m_j+2,j},\ldots,F_{i,j}) \end{align*} $$

is equal to the ideal

$$ \begin{align*} (\tilde F^{\langle m_j\rangle}_{m_j+1,j},\tilde F^{\langle m_j\rangle}_{m_j+2,j},\ldots,\tilde F^{\langle m_j\rangle}_{i,j}) \end{align*} $$

in the polynomial ring ${\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]$ . In particular, if $h:[n] \to [n]$ is an indecomposable Hessenberg function such that $h \neq (n,\ldots ,n)$ , then we have

(9.3) $$ \begin{align} {\mathbb{C}}[\operatorname{\mathrm{Hess}}(N,h) \cap \Omega_e^\circ] \cong {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(\tilde F^{\langle h(j) \rangle}_{i,j} \mid j \in J_h \ \text{and} \ h(j) < i \leq n), \end{align} $$

where .

Proof. We prove the first statement by induction on i. The base case $i = m_j+1$ is clear since $F_{m_j+1,j}=\tilde F^{\langle m_j\rangle }_{m_j+1,j}$ by equation (6.4). We proceed to the inductive step. Suppose that $i> m_j+1$ and that the claim holds for $i-1$ . Then we have

$$ \begin{align*} (F_{m_j+1,j},\ldots,F_{i-1,j},F_{i,j}) &=(F_{m_j+1,j},\ldots,F_{i-1,j}, \tilde F^{\langle m_j\rangle}_{i,j}) \ \ \ \text{(from Lemma~}9.1) \\ &=(\tilde F^{\langle m_j\rangle}_{m_j+1,j},\ldots,\tilde F^{\langle m_j\rangle}_{i-1,j},\tilde F^{\langle m_j\rangle}_{i,j}) \ \ \ \text{(by the inductive assumption)} \end{align*} $$

as desired. The isomorphism (9.3) follows from equation (3.8) and the former statement by setting $m_j=h(j)$ .

Example 9.3. Let $m_j=j+1$ . Then one has

(9.4) $$ \begin{align} &\tilde F^{\langle 2 \rangle}_{i,1}=\left| \begin{array}{@{\,}ccc@{\,}} 1 & 0 & x_{21} \\ x_{21} & 1 & x_{31} \\ x_{i1} & x_{i2} & x_{i+1 \, 1} \end{array} \right| \ \text{for} \ i> 2; \\ &\tilde F^{\langle j+1 \rangle}_{i,j}=\left| \begin{array}{@{\,}cccc@{\,}} 1 & 0 & 0 & 1 \\ x_{j \, j-1} & 1 & 0 & x_{j+1 \, j} \\ x_{j+1 \, j-1} & x_{j+1 \, j} & 1 & x_{j+2 \, j} \\ x_{i \, j-1} & x_{ij} & x_{i \, j+1} & x_{i+1 \, j} \end{array} \right| \ \text{for} \ 2 \leq j \leq n-2 \ \text{and} \ i > j+1. \notag \end{align} $$

By equation (9.3), the coordinate ring ${\mathbb {C}}[\operatorname {\mathrm {Pet}}_n \cap \Omega _e^\circ ]$ is

(9.5) $$ \begin{align} {\mathbb{C}}[\operatorname{\mathrm{Pet}}_n \cap \Omega_e^\circ] \cong {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(\tilde F^{\langle j+1 \rangle}_{i,j} \ \text{for all} \ 1 \leq j \leq n-2 \ \text{and} \ j+1 < i \leq n). \end{align} $$

We now explain the cyclic quotient singularity which is also called the type A-singularity. Let $\zeta $ be a primitive n-th root of unity and $\mathfrak C_n$ the cyclic group of order n generated by $\zeta $ . Consider the action of $\mathfrak C_n$ on ${\mathbb {C}}^2$ defined by $\zeta \cdot (x,y)=(\zeta x, \zeta ^{-1} y)$ for $\zeta \in \mathfrak C_n$ and $(x,y) \in {\mathbb {C}}^2$ . This induces the action of $\mathfrak C_n$ on the polynomial ring ${\mathbb {C}}[x,y]$ (which is the coordinate ring of ${\mathbb {C}}^2$ ), and it is given by $\zeta \cdot x=\zeta x$ and $\zeta \cdot y=\zeta ^{-1} y$ . As is well known, the $\mathfrak C_n$ -invariants in ${\mathbb {C}}[x,y]$ is isomorphic to

$$ \begin{align*} {\mathbb{C}}[{\mathbb{C}}^2/\mathfrak C_n] \cong {\mathbb{C}}[x,y]^{\mathfrak C_n} \cong {\mathbb{C}}[X,Y,Z]/(XY-Z^n), \end{align*} $$

which sends X to $x^n$ , Y to $y^n$ , and Z to $xy$ . The quotient space ${\mathbb {C}}^2/\mathfrak C_n$ is called the cyclic quotient singularity or the type $A_{n-1}$ -singularity.

Example 9.4. Let $n=3$ . As seen in Example 9.3, the coordinate ring of $\operatorname {\mathrm {Pet}}_3 \cap \Omega _e^\circ $ is given by

$$ \begin{align*} {\mathbb{C}}[\operatorname{\mathrm{Pet}}_3 \cap \Omega_e^\circ] \cong {\mathbb{C}}[x_{21}, x_{31}, x_{32}]/(\tilde F^{\langle 2 \rangle}_{3,1}). \end{align*} $$

Here, one can compute $\tilde F^{\langle 2 \rangle }_{3,1}$ as

$$ \begin{align*} \tilde F^{\langle 2 \rangle}_{3,1} &=\left| \begin{array}{@{\,}ccc@{\,}} 1 & 0 & x_{21} \\ x_{21} & 1 & x_{31} \\ x_{31} & x_{32} & 0 \end{array} \right| =x_{21}^2x_{32}-x_{21}x_{31}-x_{31}x_{32} \\ &=-x_{21}^3+(x_{21}^2-x_{31})(x_{21}+x_{32}). \end{align*} $$

Thus, we have

$$ \begin{align*} {\mathbb{C}}[\operatorname{\mathrm{Pet}}_3 \cap \Omega_e^\circ] \cong {\mathbb{C}}[X,Y,Z]/(XY-Z^3), \end{align*} $$

which sends X to $x_{21}^2-x_{31}$ , Y to $x_{21}+x_{32}$ and Z to $x_{21}$ . The ring isomorphism above yields the isomorphism $\operatorname {\mathrm {Pet}}_3 \cap \Omega _e^\circ \cong {\mathbb {C}}^2/\mathfrak C_3$ .

In Example 9.4, we constructed the polynomial $XY-Z^n$ from $\tilde F^{\langle 2 \rangle }_{3,1}$ for $n=3$ . We now generalize this fact to arbitrary n. More specifically, we construct the polynomial $XY-Z^n$ from $\tilde F^{\langle 2 \rangle }_{i,1} \ (2 < i \leq n)$ defined in equation (9.4). By the cofactor expansion along the second column, we have

(9.6) $$ \begin{align} &\tilde F^{\langle 2 \rangle}_{i,1}=\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{i1} & x_{i+1 \, 1} \end{array} \right| - x_{i2}\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{21} & x_{31} \end{array} \right| \ \text{for} \ i> 2. \end{align} $$

Lemma 9.5. For $2 < i \leq n$ , we have

$$ \begin{align*} &\tilde F^{\langle 2 \rangle}_{n,1}+x_{21}\tilde F^{\langle 2 \rangle}_{n-1,1}+x_{21}^2\tilde F^{\langle 2 \rangle}_{n-2,1}+\cdots+x_{21}^{n-i} \tilde F^{\langle 2 \rangle}_{i,1} \\ =& \left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21}^{n-i+1} \\ x_{i1} & 0 \end{array} \right| - (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-i}x_{i2})\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{21} & x_{31} \end{array} \right|. \end{align*} $$

Proof. We prove this by descending induction on i. The base case $i=n$ is the equality (9.6) for $i=n$ . Suppose now that $i < n$ and that the claim holds for $i+1$ . Then we have

$$ \begin{align*} &\tilde F^{\langle 2 \rangle}_{n,1}+x_{21}\tilde F^{\langle 2 \rangle}_{n-1,1}+x_{21}^2\tilde F^{\langle 2 \rangle}_{n-2,1}+\cdots+x_{21}^{n-i-1} \tilde F^{\langle 2 \rangle}_{i+1,1}+x_{21}^{n-i} \tilde F^{\langle 2 \rangle}_{i,1} \\ =& \left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21}^{n-i} \\ x_{i+1 \, 1} & 0 \end{array} \right| - (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-i-1}x_{i+1 \, 2})\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{21} & x_{31} \end{array} \right| + x_{21}^{n-i} \tilde F^{\langle 2 \rangle}_{i,1} \\ & \hspace{80pt} \text{(by our descending induction hypothesis)} \\ =& \left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21}^{n-i} \\ x_{i+1 \, 1} & 0 \end{array} \right| - (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-i-1}x_{i+1 \, 2})\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{21} & x_{31} \end{array} \right| \\ &+ x_{21}^{n-i} \left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{i1} & x_{i+1 \, 1} \end{array} \right| - x_{21}^{n-i} x_{i2}\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{21} & x_{31} \end{array} \right| \hspace{30pt} \text{(by equation }(9.6)) \\ =& \left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21}^{n-i+1} \\ x_{i1} & 0 \end{array} \right| - (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-i}x_{i2})\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{21} & x_{31} \end{array} \right| \end{align*} $$

as desired.

Proposition 9.6. We set

$$ \begin{align*} \begin{cases} X= x_{21}^2-x_{31} \\ Y= x_{21}^{n-2}+x_{32}x_{21}^{n-3}+\cdots+x_{n-1 \, 2}x_{21}+x_{n2} \\ Z= x_{21}. \end{cases} \end{align*} $$

Then one can write

$$ \begin{align*} \tilde F^{\langle 2 \rangle}_{n,1}+x_{21}\tilde F^{\langle 2 \rangle}_{n-1,1}+x_{21}^2\tilde F^{\langle 2 \rangle}_{n-2,1}+\cdots+x_{21}^{n-3} \tilde F^{\langle 2 \rangle}_{3,1} = XY-Z^n. \end{align*} $$

Proof. By using Lemma 9.5 for $i=3$ , we obtain

$$ \begin{align*} &\tilde F^{\langle 2 \rangle}_{n,1}+x_{21}\tilde F^{\langle 2 \rangle}_{n-1,1}+x_{21}^2\tilde F^{\langle 2 \rangle}_{n-2,1}+\cdots+x_{21}^{n-3} \tilde F^{\langle 2 \rangle}_{3,1} \\ =& \left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21}^{n-2} \\ x_{31} & 0 \end{array} \right| - (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-3}x_{32})\left| \begin{array}{@{\,}cc@{\,}} 1 & x_{21} \\ x_{21} & x_{31} \end{array} \right| \\ =& -x_{21}^{n-2}x_{31} - (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-3}x_{32})(x_{31}-x_{21}^2) \\ =& -x_{21}^n+x_{21}^{n-2}(x_{21}^2-x_{31}) + (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-3}x_{32})(x_{21}^2-x_{31})\\ =& -x_{21}^n + (x_{n2}+x_{21}x_{n-1 \, 2}+x_{21}^2x_{n-2 \, 2}+\cdots+x_{21}^{n-3}x_{32}+x_{21}^{n-2})(x_{21}^2-x_{31})\\ =&XY-Z^n.\\[-39pt] \end{align*} $$

Theorem 9.7. If $h_2=(2,n,\ldots ,n)$ , then there is an isomorphism of ${\mathbb {C}}$ -algebras

$$ \begin{align*} {\mathbb{C}}[\operatorname{\mathrm{Hess}}(N,h_2) \cap \Omega_e^\circ] \cong \frac{{\mathbb{C}}[X,Y,Z]}{(XY-Z^n)} \otimes {\mathbb{C}}[x_{32},x_{42},\ldots, x_{n-1 \, 2}] \otimes {\mathbb{C}}[x_{ij} \mid 3 \leq j < i \leq n] \end{align*} $$

which sends

(9.7) $$ \begin{align} \begin{array}{l} X \mapsto x_{21}^2-x_{31}; \\ Y \mapsto x_{21}^{n-2}+x_{32}x_{21}^{n-3}+\cdots+x_{n-1 \, 2}x_{21}+x_{n2}; \\ Z \mapsto x_{21}; \\ x_{i2} \mapsto x_{i2} \ \mathrm{for} \ 3 \leq i \leq n-1; \\ x_{ij} \mapsto x_{ij} \ \mathrm{for} \ 3 \leq j < i \leq n. \end{array} \end{align} $$

In other words,

$$ \begin{align*} \operatorname{\mathrm{Hess}}(N,h_2) \cap \Omega_e^\circ \cong {\mathbb{C}}^2/\mathfrak C_n \times {\mathbb{C}}^{\frac{1}{2}(n-1)(n-2)-1}. \end{align*} $$

Proof of Theorem 9.7.

By equation (9.3), we have

(9.8) $$ \begin{align} {\mathbb{C}}[\operatorname{\mathrm{Hess}}(N,h_2) \cap \Omega_e^\circ] \cong {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(\tilde F^{\langle 2 \rangle}_{3,1}, \tilde F^{\langle 2 \rangle}_{4,1}, \ldots, \tilde F^{\langle 2 \rangle}_{n,1}). \end{align} $$

Put

It then follows from Proposition 9.6 that

(9.9) $$ \begin{align} {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(\tilde F^{\langle 2 \rangle}_{3,1}, \ldots, \tilde F^{\langle 2 \rangle}_{n-1,1}, \tilde F^{\langle 2 \rangle}_{n,1}) \cong {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(\tilde F^{\langle 2 \rangle}_{3,1}, \ldots, \tilde F^{\langle 2 \rangle}_{n-1,1}, P_n). \end{align} $$

By the definition (9.4), $\tilde F^{\langle 2 \rangle }_{i,1}=0$ if and only if

$$ \begin{align*} x_{i+1 \, 1}=x_{21}x_{i1}+x_{31}x_{i2}-x_{21}^2x_{i2} \end{align*} $$

for $3 \leq i \leq n-1$ . These equalities for $i=n-1,n-2,\ldots ,3$ lead us to the isomorphism

(9.10) $$ \begin{align} {\mathbb{C}}[x_{ij} \mid 1 \leq j < i \leq n]/(\tilde F^{\langle 2 \rangle}_{3,1}, \ldots, \tilde F^{\langle 2 \rangle}_{n-1,1}, P_n) \cong {\mathbb{C}}[x_{21},x_{31},x_{ij} \mid 2 \leq j < i \leq n]/(P_n). \end{align} $$

It is straightforward to see that

(9.11) $$ \begin{align} & {\mathbb{C}}[x_{21},x_{31},x_{ij} \mid 2 \leq j < i \leq n]/(P_n) \\ \cong &{\mathbb{C}}[X,Y,Z]/(XY-Z^n) \otimes {\mathbb{C}}[x_{32},x_{42},\ldots, x_{n-1 \, 2}] \otimes {\mathbb{C}}[x_{ij} \mid 3 \leq j < i \leq n], \notag \end{align} $$

which sends

$$ \begin{align*} \begin{array}{l} X \mapsto x_{21}^2-x_{31}; \\ Y \mapsto x_{21}^{n-2}+x_{32}x_{21}^{n-3}+\cdots+x_{n-1 \, 2}x_{21}+x_{n2}; \\ Z \mapsto x_{21}; \\ x_{i2} \mapsto x_{i2} \ \mathrm{for} \ 3 \leq i \leq n-1; \\ x_{ij} \mapsto x_{ij} \ \mathrm{for} \ 3 \leq j < i \leq n. \end{array} \end{align*} $$

In fact, the inverse map is given by

$$ \begin{align*} \begin{array}{l} x_{21} \mapsto Z; \\ x_{31} \mapsto -X+Z^2; \\ x_{n2} \mapsto Y-Z^{n-2}-x_{32}Z^{n-3}-\cdots-x_{n-1 \, 2}Z; \\ x_{i2} \mapsto x_{i2} \ \mathrm{for} \ 3 \leq i \leq n-1; \\ x_{ij} \mapsto x_{ij} \ \mathrm{for} \ 3 \leq j < i \leq n. \end{array} \end{align*} $$

Combining equations (9.8), (9.9), (9.10) and (9.11), we conclude that

$$ \begin{align*} {\mathbb{C}}[\operatorname{\mathrm{Hess}}(N,h_2) \cap \Omega_e^\circ] \cong \frac{{\mathbb{C}}[X,Y,Z]}{(XY-Z^n)} \otimes {\mathbb{C}}[x_{32},x_{42},\ldots, x_{n-1 \, 2}] \otimes {\mathbb{C}}[x_{ij} \mid 3 \leq j < i \leq n]. \end{align*} $$

We complete the proof.

For a polynomial $f \in {\mathbb {C}}[\Omega _e^\circ ] \cong {\mathbb {C}}[x_{ij} \mid 1 \leq j < i \leq n]$ , we denote the zero set of f by

(9.12)

Corollary 9.9. The singular locus of $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ is given by

$$ \begin{align*} \operatorname{\mathrm{Sing}}(\operatorname{\mathrm{Hess}}(N,h_2) \cap \Omega_e^\circ) = \bigcap_{i=2}^n V(x_{i1}) \cap V(x_{n2}). \end{align*} $$

Proof. By Theorem 9.7, we have

$$ \begin{align*} \operatorname{\mathrm{Hess}}(N,h_2) \cap \Omega_e^\circ \cong {\mathbb{C}}^2/\mathfrak C_n \times {\mathbb{C}}^{\frac{1}{2}(n-1)(n-2)-1} \cong V(XY-Z^n). \end{align*} $$

Here, $V(XY-Z^n)$ denotes the hypersurface defined by the equation $XY-Z^n=0$ in ${\mathbb {C}}^{\frac {1}{2}(n-1)(n-2)+2}$ with the variables $X,Y,Z,x_{32},x_{42},\ldots , x_{n-1 \, 2}, x_{ij} \ (3 \leq j < i \leq n)$ . The singular locus of $V(XY-Z^n)$ is the solution set of the equations

$$ \begin{align*} X = 0, \ Y=0, \ Z=0. \end{align*} $$

It follows from the correspondence (9.7) that the image of the ideal $(X,Y,Z)$ under the isomorphism ${\mathbb {C}}[\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ ] \cong \frac {{\mathbb {C}}[X,Y,Z]}{(XY-Z^n)} \otimes {\mathbb {C}}[x_{32},x_{42},\ldots , x_{n-1 \, 2}] \otimes {\mathbb {C}}[x_{ij} \mid 3 \leq j < i \leq n]$ is

$$ \begin{align*} (x_{21}^2-x_{31}, x_{21}^{n-2}+x_{32}x_{21}^{n-3}+\cdots+x_{n-1 \, 2}x_{21}+x_{n2}, x_{21}) = (x_{31}, x_{n2}, x_{21}) \end{align*} $$

in ${\mathbb {C}}[\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ ]$ . Hence, by the presentation (9.3), the singular locus of $\operatorname {\mathrm {Hess}}(N,h_2)$ is the solution set of the equations

(9.13) $$ \begin{align} \begin{cases} x_{21}=0 \\ x_{31}=0 \\ x_{n2}=0 \\ \tilde F^{\langle 2 \rangle}_{i,1}=0 \ \ \ \text{for} \ 3 \leq i \leq n-1. \end{cases} \end{align} $$

Note that the equation $\tilde F^{\langle 2 \rangle }_{n,1}=0$ can be removed above since $\tilde F^{\langle 2 \rangle }_{n,1}=0$ is derived from the equations (9.13) by Proposition 9.6. It then follows from the equality $\tilde F^{\langle 2 \rangle }_{i,1} = x_{i+1 \, 1}+x_{21}^2x_{i2}-x_{21}x_{i1}-x_{31}x_{i2}$ for $3 \leq i \leq n-1$ that the solution set of the equations (9.13) is given by $x_{i1}=x_{n2}=0$ for $2 \leq i \leq n$ , as desired.

10 Singular locus of $\operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ $

We now give an explicit description for the singular locus of $\operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ $ where $h_m$ is defined in equation (8.7).

Theorem 10.1. Let $2 \leq m \leq n-1$ . Then, the singular locus of $\operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ $ is described as

$$ \begin{align*} \operatorname{\mathrm{Sing}}(\operatorname{\mathrm{Hess}}(N,h_m) \cap \Omega_e^\circ) = \bigcap_{i=2}^n V(x_{i1}) \cap \bigcap_{j=2}^{m} V(x_{nj}), \end{align*} $$

where the notation $V(f)$ is defined in equation (9.12).

Proof. We prove this by induction on m. The base case is $m=2$ , which follows from Corollary 9.9. Suppose now that $m> 2$ and that the claim holds for $m-1$ . It follows from Proposition 8.5 that the singular locus of $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ is given by the solution set of the equations

(10.1) $$ \begin{align} \frac{\partial}{\partial q_{rs}} \, ^{h_m} E_{n}^{(n)} = 0 \ \ \ \text{for all} \ 2 \leq s \leq n \ \text{and} \ n-h_m(n+1-s)<r \leq s. \end{align} $$

By Lemma 8.2, we have

(10.2) $$ \begin{align} \begin{cases} E_1^{[n,n]} &= x_n \\ E_2^{[n-1,n]} &= E_{1}^{[n-1,n-1]} x_n + q_{n-1 \, n} \\ E_3^{[n-2,n]} &= E_{2}^{[n-2,n-1]} x_n + E_{1}^{[n-2,n-2]} q_{n-1 \, n} + q_{n-2 \, n} \\ &\vdots \\ E_{m}^{[n-m+1,n]} &= E_{m-1}^{[n-m+1,n-1]} x_n + \cdots + E_1^{[n-m+1,n-m+1]} q_{n-m+2 \, n} + q_{n-m+1 \, n}. \end{cases} \end{align} $$

Since it holds that

$$ \begin{align*} ^{h_m} E_i^{[n-i+1,n]}=\frac{\partial}{\partial q_{1 \, n-i}} \, ^{h_m} E_{n}^{(n)} = 0 \ \ \ \text{for all} \ 1 \leq i \leq m \end{align*} $$

by equations (8.6) and (10.1), the equalities (10.2) lead us to the following equation

which yields that $x_n = q_{n-1 \, n} = q_{n-2 \, n} = \cdots = q_{n-m+1 \, n} = 0$ . Since we have $q_{n-m+1 \, n} = 0$ and

$$ \begin{align*} \left. \left( \frac{\partial}{\partial q_{rs}} \, ^{h_m} E_{n}^{(n)} \right) \right|_{q_{n-m+1 \, n}=0} = \frac{\partial}{\partial q_{rs}} \, ^{h_{m-1}}E_{n}^{(n)} \end{align*} $$

for $2 \leq s \leq n$ and $n-h_{m-1}(n+1-s)<r \leq s$ , the equations (10.1) imply that

(10.3) $$ \begin{align} \begin{cases} \frac{\partial}{\partial q_{rs}} \, ^{h_{m-1}}E_{n}^{(n)} &= 0 \ \ \ \text{for all} \ 2 \leq s \leq n \ \text{and} \ n-h_{m-1}(n+1-s)<r \leq s \\ \frac{\partial}{\partial q_{n-m+1 \, n}} \, ^{h_m} E_{n}^{(n)} &= 0 \\ q_{n-m+1 \, n} &= 0. \end{cases} \end{align} $$

Conversely, the equations (10.3) yield equation (10.1). In fact, since one can write $\, ^{h_{m}}E_{n}^{(n)} = \, ^{h_{m-1}}E_{n}^{(n)}+q_{n-m+1 \, n} \cdot F$ for some polynomial F by Definition 4.10, we have $\frac {\partial }{\partial q_{rs}} \, ^{h_{m}}E_{n}^{(n)} = \frac {\partial }{\partial q_{rs}} \, ^{h_{m-1}}E_{n}^{(n)}+q_{n-m+1 \, n} \cdot \frac {\partial }{\partial q_{rs}} F$ for all $2 \leq s \leq n$ and $n-h_{m-1}(n+1-s)<r \leq s$ . Hence, equation (10.1) is equivalent to equation (10.3). One has

$$ \begin{align*} \frac{\partial}{\partial q_{n-m+1 \, n}} \, ^{h_m} E_{n}^{(n)} = \, ^{h_m} E_{n-m}^{[1,n-m]} = \, ^{h_m} E_{n-m}^{(n-m)} \end{align*} $$

from equations (8.1) and (8.6), so the singular locus of $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ is given by the solution set of the equations

$$ \begin{align*} \begin{cases}\frac{\partial}{\partial q_{rs}} \, ^{h_{m-1}}E_{n}^{(n)} &= 0 \ \ \ \text{for all} \ 2 \leq s \leq n \ \text{and} \ n-h_{m-1}(n+1-s)<r \leq s \\^{h_m} E_{n-m}^{(n-m)} &= 0 \\q_{n-m+1 \, n} &= 0.\end{cases} \end{align*} $$

By Corollary 7.4, the following commutative diagram holds

(10.4)

where both vertial arrows are surjective. The ideal defining the singular locus of the zero set $V\big (\, ^{h_m} E_1^{(n)}, \ldots , \, ^{h_m} E_n^{(n)}\big )$ is

$$ \begin{align*} \left(\frac{\partial}{\partial q_{rs}} \, ^{h_{m-1}}E_{n}^{(n)} \middle| \ 2 \leq s \leq n, n-h_{m-1}(n+1-s)<r \leq s \right) + (^{h_m} E_{n-m}^{(n-m)})+(q_{n-m+1 \, n}). \end{align*} $$

The image of the ideal above under the isomorphism $\varphi _{h_m}^{-1}$ in equation (10.4) is equal to

$$ \begin{align*} \left(\varphi^{-1}\left(\frac{\partial}{\partial q_{rs}} \, ^{h_{m-1}}E_{n}^{(n)}\right) \middle| \ 2 \leq s \leq n, n-h_{m-1}(n+1-s)<r \leq s \right) + (x_{nm})+(F_{m,1}) \end{align*} $$

in ${\mathbb {C}}[\operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ ]$ by the definition (7.1) and Proposition 7.3. Hence, the singular locus of $\operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ $ is the solution set of the equations

(10.5) $$ \begin{align} \begin{cases} \varphi^{-1}\left(\frac{\partial}{\partial q_{rs}} \, ^{h_{m-1}}E_{n}^{(n)} \right) = 0 \ \ \ \text{for all} \ 2 \leq s \leq n \ \text{and} \ n-h_{m-1}(n+1-s)<r \leq s \\ F_{m,1} = 0 \\ F_{i,1} = 0 \ \ \ \text{for all} \ m+1 \leq i \leq n \end{cases} \end{align} $$

and

$$ \begin{align*} x_{nm} = 0 \end{align*} $$

by Lemma 2.3. Since the singular locus of $\operatorname {\mathrm {Hess}}(N,h_{m-1}) \cap \Omega _e^\circ $ is the solution set of the equations (10.5) from Proposition 8.5, Theorem 4.13 and Lemma 2.3 again, we can describe

$$ \begin{align*} \operatorname{\mathrm{Sing}}(\operatorname{\mathrm{Hess}}(N,h_m) \cap \Omega_e^\circ) = \operatorname{\mathrm{Sing}}(\operatorname{\mathrm{Hess}}(N,h_{m-1}) \cap \Omega_e^\circ) \cap V(x_{nm}). \end{align*} $$

One can see from our inductive assumption that $\operatorname {\mathrm {Sing}}(\operatorname {\mathrm {Hess}}(N,h_{m-1}) \cap \Omega _e^\circ )= \bigcap _{i=2}^n V(x_{i1}) \cap \bigcap _{j=2}^{m-1} V(x_{nj})$ , so we complete the proof.

Let $\mathfrak S_n$ be the permutation group on $[n]$ . For $w \in \mathfrak S_n$ , the Schubert cell $X_w^\circ $ is defined to be the B-orbit of the permutation flag $wB$ in the flag variety $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ . The Schubert variety $X_w$ is defined by the closure of the Schubert cell $X_w^\circ $ , that is, $X_w=\overline {BwB/B}$ . We put for $i \in [n]$ , where $e_1,\ldots , e_n$ are the standard basis for ${\mathbb {C}}^n$ . Under the identification $Fl({\mathbb {C}}^n) \cong \operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ , one can describe the Schubert variety $X_w$ as

$$ \begin{align*} X_w=\{V_\bullet \in Fl({\mathbb{C}}^n) \mid \dim(V_p \cap F_q) \geq r_w(p,q) \ \text{for all} \ p,q \in [n] \}, \end{align*} $$

where $r_w(p,q)=|\{i \in [p] \mid w(i) \leq q \}|$ (e.g., [Reference Fulton19, Section 10.5]). For $2 \leq m \leq n-1$ , we define the permutation $w_m \in \mathfrak S_n$ as

(10.6) $$ \begin{align} w_m(i)=\begin{cases} 1 \ &\mathrm{if} \ i=1, \\ n+1-i \ &\mathrm{if} \ 2 \leq i \leq m, \\ n \ &\mathrm{if} \ i=m+1, \\ n+2-i \ &\mathrm{if} \ m+2 \leq i \leq n. \end{cases} \end{align} $$

Then one can verify from [Reference Gasharov and Reiner21] that the Schubert variety $X_{w_m}$ is described as

$$ \begin{align*} X_{w_m} = \{V_\bullet \in Fl({\mathbb{C}}^n) \mid \dim(V_1 \cap F_1) \geq 1 \ \text{and} \ \dim(V_m \cap F_{n-1}) \geq m \}. \end{align*} $$

In other words, $V_\bullet \in X_{w_m}$ if and only if $V_1 = F_1=\operatorname {\mathrm {span}}_{\mathbb {C}} \{e_1 \}$ and $V_m \subset F_{n-1}=\operatorname {\mathrm {span}}_{\mathbb {C}} \{e_1,\ldots ,e_{n-1} \}$ . In particular, we have

$$ \begin{align*} X_{w_m} \cap \Omega_e^\circ = \bigcap_{i=2}^n V(x_{i1}) \cap \bigcap_{j=2}^{m} V(x_{nj}). \end{align*} $$

Hence, we obtain the following result from Theorem 10.1.

Corollary 10.2. Let $2 \leq m \leq n-1$ . Then, the singular locus of $\operatorname {\mathrm {Hess}}(N,h_m) \cap \Omega _e^\circ $ is equal to

$$ \begin{align*} \operatorname{\mathrm{Sing}}(\operatorname{\mathrm{Hess}}(N,h_m) \cap \Omega_e^\circ) = X_{w_m} \cap \Omega_e^\circ. \end{align*} $$

Remark 10.3. Let A be a nilpotent matrix (not necessarily regular nilpotent). Then it is known from [Reference Escobar, Precup and Shareshian15, Theorem 5] that the singular locus of $\operatorname {\mathrm {Hess}}(A,h_{n-1})$ is

$$ \begin{align*} \operatorname{\mathrm{Sing}}(\operatorname{\mathrm{Hess}}(A,h_{n-1})) = \operatorname{\mathrm{Hess}}(A,h=(1,n-1,\ldots,n-1,n)). \end{align*} $$

Consider the regular nilpotent case. The Hessenberg function $h=(1,n-1,\ldots ,n-1,n)$ is decomposable, so every flag $V_\bullet \in \operatorname {\mathrm {Hess}}(N,h)$ has $V_{1} = \operatorname {\mathrm {span}}_{\mathbb {C}} \{e_1\}$ and $V_{n-1} = \operatorname {\mathrm {span}}_{\mathbb {C}} \{e_1,\ldots ,e_{n-1}\}$ (see Definition 3.2 and surrounding discussion). Hence, the result of [Reference Escobar, Precup and Shareshian15, Theorem 5] for $A=N$ gives

$$ \begin{align*} \operatorname{\mathrm{Sing}}(\operatorname{\mathrm{Hess}}(N,h_{n-1})) = X_{w_{n-1}}. \end{align*} $$