Cohomology ring of the flag variety vs Chow cohomology ring of the Gelfand-Zetlin toric variety

TL;DR

This paper compares the cohomology ring of the flag variety with the Chow cohomology ring of the Gelfand-Zetlin toric variety, revealing structural differences and computing specific cases.

Contribution

It demonstrates that the flag variety's cohomology ring is a Gorenstein quotient of a subalgebra of the Gelfand-Zetlin toric variety's Chow ring, providing explicit computations for small cases.

Findings

The cohomology ring of the flag variety is a Gorenstein quotient.

The subalgebra generated by degree 1 elements often lacks Poincare duality.

Explicit calculations for n=3 illustrate the general structure.

Abstract

We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.

Equations

$$F=(\{0\}⫋F_1⫋\cdots ⫋F_n={\mathbb{C}}^n).$$

$$\begin{array}[]{ccccccccc}\lambda_{1}&&\lambda_{2}&&\lambda_{3}&&\ldots&&\lambda_{n}\\ &x_{11}&&x_{12}&&\ldots&&x_{1(n-1)}&\\ &&x_{21}&&x_{22}&&\ldots&&\\ &&&\ddots&&\udots&&&\\ &&&&x_{(n-1)1}&&&&\end{array}$$

$$-1\le x\le 0,0\le y\le 1,x\le z\le y.$$

$$\dim (V_{\lambda })=\#({\Delta }_{\lambda }\cap {\mathbb{Z}}^N).$$

$$\dim (H^0(\mathcal{F}{\ell }_n,{\mathcal{L}}_{\lambda }))=\#({\Delta }_{\lambda }\cap {\mathbb{Z}}^N).$$

$$a_j$$

$$y_{(i-1)j}+a_{ij}$$

$${\lambda }_1\le x_{11}=y_{11}\le {\lambda }_2\le x_{12}\le \dots \le x_{1j}\le {\lambda }_{j+1},$$

$$\begin{array}[]{ccc}&x_{(i-2)(j+1)}&\\ x_{(i-1)j}&&x_{(i-1)(j+1)}\\ &x_{ij}&\end{array}$$

$$x_{(i-1)j}+a_{ij}^{\prime }\le y_{ij}\le x_{(i-1)(j+1)}+b_{ij}^{\prime }$$

$$x_{(i-2)(j+1)}+a_{ij}^{\prime }=x_{(i-2)(j+1)}+b_{ij}^{\prime }$$

$$x_{(i-1)j}\le x_{ij}\le x_{(i-1)(j+1)}$$

$$P+Q=\{x+y|x\in P,y\in Q\}.$$

$${\Delta }_{\lambda +\mu }={\Delta }_{\lambda }+{\Delta }_{\mu },$$

$${\lambda }_1+{\mu }_1\le x_{11}\le {\lambda }_2+{\mu }_2\le \cdots \le x_{1(n-1)}\le {\lambda }_n+{\mu }_n.$$

$$\text{deg}(X)=\#(X\cap H_1\cap \dots \cap H_d),$$

$$\text{deg}(X,\mathcal{L})=d!\underset{m\to \infty }{\lim }\frac{\dim H^0(X,{\mathcal{L}}^{\otimes m})}{m^d}.$$

$$\mathrm{deg}(X_{\Sigma },{\mathcal{L}}_P)=d!{\mathrm{Vol}}_d(P).$$

$$\text{deg}(X_{\Sigma },{\mathcal{L}}_P)=d!\underset{m\to \infty }{\lim }\frac{\dim H^0(X_{\Sigma },{\mathcal{L}}_P^{\otimes m})}{m^d}=d!\underset{m\to \infty }{\lim }\frac{\#(mP\cap {\mathbb{Z}}^d)}{m^d}=d!{\mathrm{Vol}}_d(P).$$

$${\mathcal{L}}_{\lambda }^{\otimes m}={\mathcal{L}}_{m\lambda }.$$

$$\mathrm{deg}(\mathcal{F}{\ell }_n,{\mathcal{L}}_{\lambda })=N!{\mathrm{Vol}}_N({\Delta }_{\lambda }),$$

$$\text{deg}(\mathcal{F}{\ell }_n,{\mathcal{L}}_{\lambda })=N!\underset{m\to \infty }{\lim }\frac{\dim H^0(\mathcal{F}{\ell }_n,{\mathcal{L}}_{\lambda }^{\otimes m})}{m^N}=N!\underset{m\to \infty }{\lim }\frac{\#(m{\Delta }_{\lambda }\cap {\mathbb{Z}}^N)}{m^N}=N!{\mathrm{Vol}}_N({\Delta }_{\lambda }),$$

$$H^{\ast }(\mathcal{F}{\ell }_n,\mathbb{Q})\cong \text{Sym}({\Lambda }_{\mathbb{Q}})/I$$

$$A^i\times A^{n-i}\to A^n\cong \mathbf{k}$$

$$x\cdot A^{n-\mathrm{deg}(x)}=0.$$

$$P(x_1,\dots ,x_r)=(x_1{a}_1+\cdots +x_r{a}_r{)}^n\in A^n\cong \mathbf{k}.$$

$$A\cong \mathbf{k}[{\partial }_1,\dots ,{\partial }_r]/I$$

$$a_1\cdots a_n=\phi (a_1)\cdots \phi (a_n)$$

$$A^1\stackrel{\phi }{\to }{B}^1\stackrel{q}{\to }\text{PD}(B{)}^1,$$

$$\sum _{\sigma \in \Sigma (n-k),\sigma \supset \tau }⟨u,n_{\sigma ,\tau }⟩c(\sigma )=0,\forall u\in M(\tau ):=M\cap {\tau }^{\perp }.$$

$$(c\cup \tilde{c})(\gamma )=\sum _{(\sigma ,\tau )\in \Sigma (n-p)\times \Sigma (n-q)}{m}_{\sigma ,\tau }^{\gamma }c(\sigma )\tilde{c}(\tau ),\forall \gamma \in \Sigma (n-p-q),$$

Full text

The first author is partially supported by a National Science Foundation Grant (Grant ID: DMS-1601303).

1. 1 Some facts about Gelfand-Zetlin polytopes
2. 2 Review of degrees of line bundles on toric and flag varieties
3. 3 Review of intersection theory on toric and flag varieties
4. 4 Some algebra lemmas
5. 5 Main theorem
6. 6 Minkowski weights
7. 7 Gelfand-Zetlin example, $n=3$

Introduction

Throughout the paper, the base field is assumed to be $\mathbb{C}$. The complete flag variety $\mathcal{F}\ell_{n}$ is the variety whose points parameterize complete flags of subspaces in $\mathbb{C}^{n}$, namely:

[TABLE]

The variety $\mathcal{F}\ell_{n}$ can be identified with the homogeneous space $\textup{GL}(n,\mathbb{C})/B$ where $B$ is the subgroup of upper triangular matrices. The geometry of flag variety plays an important role in representation theory of $\textup{GL}(n,\mathbb{C})$ and combinatorics related to the permutation group. More generally there is a notion of flag variety for any reductive algebraic group $G$.

We recall that $\dim(\mathcal{F}\ell_{n})=N=n(n-1)/2$. The classes of Schubert varieties form an important $\mathbb{Z}$-basis for $H^{*}(\mathcal{F}\ell_{n},\mathbb{Z})$. Since $\mathcal{F}\ell_{n}$ has a paving by affine cells (Schubert cells), it has no odd cohomology. Moreover, $H^{*}(\mathcal{F}\ell_{n},\mathbb{Z})$ is generated by degree $2$ elements. Also its Chow ring $A^{*}(\mathcal{F}\ell_{n})$ is isomorphic to $H^{*}(\mathcal{F}\ell_{n},\mathbb{Z})$ where the isomorphism doubles the degree. The famous Borel description states that $H^{*}(\mathcal{F}\ell_{n},\mathbb{Z})$ is isomorphic to the polynomial algebra in $n$ variables quotient by the ideal generated by non-constant symmetric polynomials.

We identify the weight lattice $\Lambda=\Lambda_{\textup{GL}(n,\mathbb{C})}$ with the additive group $\mathbb{Z}^{n}$ and the semigroup of dominant weights $\Lambda^{+}=\Lambda^{+}_{\textup{GL}(n,\mathbb{C})}$ (respectively the positive Weyl chamber $\Lambda^{+}_{\mathbb{R}}$) with the collection of all increasing sequences $\lambda=(\lambda_{1}\leq\cdots\leq\lambda_{n})$ of integers (respectively real numbers). If $\lambda_{1}<\cdots<\lambda_{n}$ we call $\lambda$ a regular dominant weight. We also denote the weight lattice $\Lambda(\textup{SL}(n,\mathbb{C}))$ of $\textup{SL}(n,\mathbb{C})$ by $\Lambda^{\prime}$. It can be identified with the quotient $\Lambda/\mathbb{Z}(1,\ldots,1)$.

In their fundamental work [GZ50], Gelfand and Zetlin111Warning to the reader: several different spellings of Zetlin’s name appear in the English literature such as Tsetlin, Cetlin, Zeitlin or Tzetlin. Following Valentina Kiritchenko we use the spelling Zetlin, justified by the fact that while he was Russian his last name seems to have German origins. construct a certain vector space basis $B_{\lambda}$ for an irreducible representation $V_{\lambda}$ of $\textup{GL}(n,\mathbb{C})$ with highest weight $\lambda$, and they explicitly describe the action of $\mathfrak{gl}(n,\mathbb{C})=\textup{Lie}(\textup{GL}(n,\mathbb{C}))$ on basis elements in $B_{\lambda}$. The Gelfand-Zetlin basis $B_{\lambda}$ has the remarkable property that its elements are indexed by the lattice points in a convex polytope $\Delta_{\lambda}\subset\mathbb{R}^{N}$, where $N=n(n-1)/2$, called the Gelfand-Zetlin polytope (or GZ polytope) associated to $\lambda$. The defining inequalities of $\Delta_{\lambda}$ can be explicitly written down. If $\lambda=(\lambda_{1}\leq\cdots\leq\lambda_{n})$ the polytope $\Delta_{\lambda}$ is the collection of $(x_{ij}\mid 1\leq i\leq n-1,1\leq j\leq n-i)\in\mathbb{R}^{N}$ satisfying the following array of inequalities:

[TABLE]

where each small triangle \begin{array}[]{ccc}a&&b\\ &c&\end{array} corresponds to the inequalities $a\leq c\leq b$. For example if $\lambda=(-1,0,1)$, the Gelfand-Zetlin polytope $\Delta_{\lambda}$ is given by the inequalities (see Figure 1):

[TABLE]

Since there is a one-to-one correspondence between the elements of the Gelfand-Zetlin basis $B_{\lambda}$ and the lattice points in $\Delta_{\lambda}$ one immediately sees that:

[TABLE]

It is well-known that a weight $\lambda$ gives rise to a $\textup{GL}(n,\mathbb{C})$-linearized line bundle $\mathcal{L}_{\lambda}$ on the flag variety $\mathcal{F}\ell_{n}$. When $\lambda$ is regular dominant the line bundle $\mathcal{L}_{\lambda}$ is very ample. By the Borel-Weil theorem $H^{0}(\mathcal{F}\ell_{n},\mathcal{L}_{\lambda})\cong V_{\lambda}^{*}$ as a $\textup{GL}(n,\mathbb{C})$-module. Thus, in particular we have:

[TABLE]

A general philosophy, suggested in the work of several authors and in particular A. Okounkov [Oko98], is that GZ polytopes play a role for the flag variety similar to that of Newton polytopes for toric varieties. In this direction in [Kav11] the first author obtains a description of $H^{*}(\mathcal{F}\ell_{n},\mathbb{Q})$ in terms of volumes of GZ polytopes. This description is very similar to the Khovanskii-Pukhlikov description of cohomology ring of a smooth projective toric variety in terms of volumes of Newton polytopes. The description in [Kav11] turns out to be equivalent to the Borel description via a theorem of Kostant (see [Kav11, Remark 5.4]). Making the connection between geometry of $\mathcal{F}\ell_{n}$ and GZ polytopes stronger, in [KST12] the authors make a correspondence between Schubert varieties and certain unions of faces of GZ polytopes. They use this correspondence to give applications in Schubert calculus.

It can be shown that for regular dominant weights $\lambda$, all the polytopes $\Delta_{\lambda}$ have the same normal fan (Proposition 1.1). We call this common normal fan the Gelfand-Zetlin fan and denote it by $\Sigma_{\textup{GZ}}$. It is well-known that, for each regular dominant $\lambda$ the pair $(\mathcal{F}\ell_{n},\mathcal{L}_{\lambda})$ can be degenerated, in a flat family with reduced irreducible fibers, to $(X_{\textup{GZ}},\mathcal{L}_{\Delta_{\lambda}})$. Here $\mathcal{L}_{\Delta_{\lambda}}$ is the equivariant line bundle on the toric variety $X_{\textup{GZ}}$ corresponding to the lattice polytope $\Delta_{\lambda}$ (see [KM05]). Such degenerations have been used to study mirror symmetry for the flag variety and partial flag varieties (see [BCFKvS00]). This motivates the problem of comparing the geometry and topology of $\mathcal{F}\ell_{n}$ with that of $X_{\textup{GZ}}$.

The variety $X_{\textup{GZ}}$ is not smooth and hence its Chow group does not have a ring structure. There is a dual version of the Chow ring, due to Fulton and MacPherson [FM81], that works for singular varieties as well. It is called the operational Chow ring or simply Chow cohomology ring. For a variety $X$ we denote its Chow cohomology ring by $A^{*}(X)$.

Let ${\bf k}$ be a field. Given a graded algebra $A=\bigoplus_{i=0}^{n}A^{i}$ with $A^{0}\cong A^{n}\cong{\bf k}$, one can form the largest quotient $A/I$ of $A$ such that $A/I$ has Poincaré duality (Lemma 4.1). We call this the Poincaré duality quotient of $A$ and denote it by $\textup{PD}(A)$. The main result of the paper is the following (Theorem 5.1):

Theorem 1**.**

The cohomology ring $H^{*}(\mathcal{F}\ell_{n},\mathbb{Q})$ is isomorphic to the Poincaré duality quotient of the subalgebra of $A^{*}(X_{\textup{GZ}},\mathbb{Q})$ generated by degree $1$ elements.

One key combinatorial ingredient in the proof is the following statement suggested to us by Valentina Kiritchenko (Proposition 1.3):

Proposition 2**.**

Let $P$ be a polytope whose normal fan is $\Sigma_{GZ}$, then $P=c+\Delta_{\lambda}$ for some $\lambda\in\Lambda^{+}$ and $c\in\mathbb{R}^{N}$.

Another ingredient in the proof of Theorem 1 is an algebra lemma which states that a Poincaré duality algebra $A=\bigoplus_{i=0}^{n}A^{i}$ that is finite dimensional as a vector space and is generated (over $A^{0}$) by $A^{1}$, is uniquely determined by its top product polynomial $p:A^{1}\to A^{n}\cong A^{0}$, $p(x)=x^{n}$ (Theorem 4.2).

In [FS97] it is shown that the Chow cohomology ring of a toric variety $X_{\Sigma}$ is naturally isomorphic to the ring of Minkowski weights on its fan $\Sigma$. A degree $k$ Minkowski weight on a fan $\Sigma$ is an assignment of integers to $k$-dimensional cones in $\Sigma$ which satisfies certain balancing condition. One defines a product of Minkowski weights that makes the collection of all Minkowski weights into a ring (see Section 6, see also [FS97, Kaz03]). There is also an alternative description of the Chow cohomology ring of a toric variety in terms of piecewise linear functions on its fan (see [Pay06]).

In Section 7, we use the Minkowski weights description of the Chow cohomology ring, to compute $A^{*}(X_{\textup{GZ}},\mathbb{Q})$ for $n=3$ and see directly that it coincides with its subalgebra generated by degree $1$ elements. We also see $A^{*}(X_{\textup{GZ}},\mathbb{Q})$ does not have Poincaré duality.

The second author has written a Sage code that verifies that for $n=4,5$ the Chow cohomology ring of $X_{\textup{GZ}}$ is not generated in degree $1$, and moreover the subalgebra generated in degree $1$ does not have Poincaré duality. See https://github.com/evillella/minkowski. Also see the appendix in the second author’s Ph.D. thesis [Vil19].

In geometric terms, the isomorphism between the Picard groups of $\mathcal{F}\ell_{n}$ and $X_{\textup{GZ}}$ can be constructed by means of a toric degeneration. A toric degeneration of $\mathcal{F}\ell_{n}$ to $X_{\textup{GZ}}$ is a flat family $\pi:\mathcal{X}\to\mathbb{C}$ with reduced fibers and an action of $\mathbb{C}^{*}$ lifting the $\mathbb{C}^{*}$ action on the base $\mathbb{C}$ such that the general fiber $X_{t}:=\pi^{-1}(t)$, $t\neq 0$, is $\mathcal{F}\ell_{n}$ and its unique special fiber $X_{0}:=\pi^{-1}(0)$ is $X_{\textup{GZ}}$. Then any divisor class $[D]$ in $\operatorname{Pic}(\mathcal{F}\ell_{n})$ can be extended to the whole family $\mathcal{X}$ and then specialized to the special fiber $X_{\textup{GZ}}$ to get a divisor class $[D_{0}]$ on $X_{\textup{GZ}}$. For a general toric degeneration, $[D_{0}]$ may not be a Cartier divisor class. But one shows that this is the case for example for the family constructed in [KM05] (see [KM05, Proposition 11]). In fact, under this specialization map the class of a line bundle $\mathcal{L}_{\lambda}$ on $\mathcal{F}\ell_{n}$ goes to the class of the line bundle on $X_{\textup{GZ}}$ determined by the polytope $\Delta_{\lambda}$. We do not know if this construction extends to give a homomorphism between the Chow cohomology rings.

Acknowledgements We thank Valentina Kiritchenko and June Huh for very helpful email correspondence and conversations. We are also in debt to Kyeong-Dong Park for proof reading a first version of the paper and giving useful comments.

1. Some facts about Gelfand-Zetlin polytopes

In this section we prove some basic facts about GZ polytopes. We start with the normal fan to a GZ polytope $\Delta_{\lambda}$. Recall that the normal fan $\Sigma_{\Delta}$ of a polytope $\Delta$ is constructed as follows: for each face $F$ let $C_{F}$ be the face cone of $F$ and let $\sigma_{F}$ be the dual cone to $C_{F}$. Then $\Sigma_{\Delta}=\{\sigma_{F}\mid F\textup{ face of }\Delta\}$ (see [CLS11, Section 2.3]).

Proposition 1.1**.**

For a regular dominant weight $\lambda$, the normal fan $\Sigma_{\lambda}$ of $\Delta_{\lambda}$ is independent of $\lambda$.

Proof.

The facets of $\Delta_{\lambda}$ correspond to single equalities in the array (1), and lower dimensional faces of $\Delta_{\lambda}$ correspond to multiple equalities in the array. There are two types of equality that can occur, (i) those of the form $x_{1i}=\lambda_{j}$ and (ii) those of the form $x_{ij}=x_{(i-1)k}$. The second type of equality is clearly independent of $\lambda$ and the first type depends on $\lambda$, so that the faces corresponding to various $\lambda$ values differ only by translation. It follows that for a face $F$, which is defined by a combination of these two types of equalities, the corresponding face cone $C_{F}$ and hence its dual cone $\sigma_{F}$ is independent of $\lambda$. This proves the claim. ∎

Definition 1.2** (Gelfand-Zetlin fan).**

We refer to the common normal fan of the $\Delta_{\lambda}$, where $\lambda$ is regular dominant, as the Gelfand-Zetlin fan and denote it by $\Sigma_{\textup{GZ}}$.

Proposition 1.3**.**

Let $P$ be a polytope whose normal fan is $\Sigma_{GZ}$, then $P=c+\Delta_{\lambda}$ for some $\lambda\in\Lambda^{+}$ and $c\in\mathbb{R}^{N}$. Moreover, if $P$ is a lattice polytope then $\lambda$ is a dominant weight and $c\in\mathbb{Z}^{N}$.

Proof.

Since normal fan of $P$ is $\Sigma_{GZ}$, the hyperplanes defining $P$ are parallel to the ones defining $\Delta_{\lambda}$, for any dominant regular $\lambda$ (as we have already showed the fan is independent of $\lambda$). Let us use $y_{ij}$ (respectively $x_{ij}$) for coordinates of a point in $P$ (respectively a GZ polytope $\Delta_{\lambda}$). Recall that there are two types of inequalities defining $\Delta_{\lambda}$ namely, $\lambda_{i}\leq x_{1i}\leq\lambda_{i+1}$ and $x_{(i-1)j}\leq x_{ij}\leq x_{(i-1)(j+1)}$. Since the facets of $P$ are parallel to those of a GZ polytope we conclude that the inequalities defining $P$ come in two types as well:

[TABLE]

We wish to find $\lambda=(\lambda_{1}\leq\cdots\leq\lambda_{n})$ and $c=(c_{ij})\in\mathbb{R}^{N}$ such that if $x_{ij}=y_{ij}+c_{ij}$ then the inequalities (2) for the $y_{ij}$ are equivalent to the GZ inequalities (1) for the $x_{ij}$.

The first type of inequalities $a_{j}\leq y_{1j}\leq b_{j}$ tell us what $\lambda$ to choose. Set $\lambda_{1}=a_{1}$ and $\lambda_{2}=b_{1}$. By induction suppose for $1\leq j<n-1$ we have picked $\lambda_{1},\ldots,\lambda_{j+1}$ and $c_{11}=0,c_{12},\ldots,c_{1j}$ such that:

[TABLE]

where $x_{1k}=y_{1k}+c_{1k}$, for all $k$. Now if we put $c_{1(j+1)}=\lambda_{j+1}-a_{j+1}$ and $\lambda_{j+2}=b_{j+1}+\lambda_{j+1}-a_{j+1}$ we have $\lambda_{j+1}\leq x_{1(j+1)}\leq\lambda_{j+2}$ as required.

For the remaining rows, we first need to examine the small diamonds \begin{array}[]{ccc}&a&\\ b&&c\\ &d&\end{array} appearing in the GZ array (1). Since $b\leq d\leq c$, the equalities $b=a$ and $c=a$ imply $d=a$. This gives us linear relations among the ray generators in the fan $\Sigma_{\textup{GZ}}$ which in turn translate to relations among the $a_{ij}$, $b_{ij}$ for the polytope $P$. Let $1<i<n-1$ and $1\leq j\leq n-i$ and by induction suppose we have picked $c_{11},\ldots,c_{i(j-1)}$ so that $x_{11},\ldots,x_{i(j-1)}$ satisfy the GZ triangular array of inequalities. We would like to find $c_{ij}$ so that $x_{ij}=y_{ij}+c_{ij}$ satisfies the diamond

[TABLE]

The second type of inequalities in (2) can be written as:

[TABLE]

where $a^{\prime}_{ij}=a_{ij}+c_{(i-1)j}$ and $b^{\prime}_{ij}=b_{ij}+c_{(i-1)(j+1)}$. Now when we consider the face of $P$ where $x_{(i-1)j}=x_{(i-2)(j+1)}$ and $x_{(i-1)(j+1)}=x_{(i-2)(j+1)}$, by what we said above, the inequality (3) becomes two equalities. We thus have:

[TABLE]

which implies $a^{\prime}_{ij}=b^{\prime}_{ij}$. Now, if we define $x_{ij}=y_{ij}-a^{\prime}_{ij}$, i.e. $c_{ij}=-a^{\prime}_{ij}$, the relation (3) becomes

[TABLE]

as required. Therefore $P=c+\Delta_{\lambda}$ where $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ and $c=(c_{ij})$ as constructed above. Finally, if $P$ is a lattice polytope then the $a_{i}$, $b_{i}$, $a_{ij}$, $b_{ij}$ should be integers (note that none of the inequalities in (2) is redundant and the corresponding equality defines a facet of $P$). This implies that $\lambda$ and $c$ are integer vectors as well. ∎

Remark 1.4**.**

Proposition 1.3 was suggested to us by Valentina Kiritchenko. The proof presented above is due to the second author.

Remark 1.5**.**

Observe that there are $n+n(n-1)/2$ parameters present in $c+\Delta_{\lambda}$, but a GZ polytope is cut out by $n(n-1)$ facets, one for each ray in $\Sigma_{\textup{GZ}}(1)$. The dimension of the space of polytopes with normal fan $\Sigma_{GZ}$ is hence much smaller than the number of rays in the fan due to the fact that $\Delta_{\lambda}$ is not a simple polytope, or equivalently, the fan $\Sigma_{GZ}$ is not simplicial.

A third useful property of the GZ polytopes is that they behave well with respect to Minkowski addition. We recall that for polytopes $P$ and $Q$, the Minkowski sum $P+Q$ is the polytope

[TABLE]

Proposition 1.6**.**

The assignment $\lambda\mapsto\Delta_{\lambda}$ is additive, that is, for any dominant weights $\lambda,\mu\in\mathbb{Z}^{n}$ we have:

[TABLE]

where the addition on the right is the Minkowski sum of polytopes.

Proof.

The inclusion $\Delta_{\lambda}+\Delta_{\mu}\subset\Delta_{\lambda+\mu}$ is clear. We need to show the other direction. Let $x\in\Delta_{\lambda+\mu}$, our goal is to write $x=x^{\prime}+x^{\prime\prime}$ with $x^{\prime}\in\Delta_{\lambda}$ and $x^{\prime\prime}\in\Delta_{\mu}$. We begin with the first row $x_{1*}=(x_{11},\ldots,x_{1(n-1)})$ satisfying

[TABLE]

It is clear that, for each $i$, the sum of line segments $[\lambda_{i},\lambda_{i+1}]$ and $[\mu_{i},\mu_{i+1}]$ is $[\lambda_{i}+\mu_{i},\lambda_{i+1}+\mu_{i+1}]$. Thus we can find $x^{\prime}_{1*},x^{\prime\prime}_{1*}\in\mathbb{R}^{n-1}$ such that $x_{1*}=x^{\prime}_{1*}+x^{\prime\prime}_{1*}$ and they satisfy the first row of interlacing inequalities for $\Delta_{\lambda}$ and $\Delta_{\mu}$ respectively. We can then repeat the same argument for the second row replacing $\lambda$, $\mu$ with $x^{\prime}_{1*}$, $x^{\prime\prime}_{1*}$ to obtain $x^{\prime}_{2*},x^{\prime\prime}_{2*}\in\mathbb{R}^{n-2}$. Continuing this way we find $x^{\prime}\in\Delta_{\lambda}$, $x^{\prime\prime}\in\Delta_{\mu}$ with $x=x^{\prime}+x^{\prime\prime}$ as required. ∎

Remark 1.7**.**

Proposition 1.6 shows that the collection of Gelfand-Zetlin polytopes is an example of a linear family of polytopes (as defined in [KV18]). In this regard, Proposition 1.1 is related to [KV18, Proposition 1.3].

Proposition 1.8**.**

Suppose for two dominant weights $\lambda$, $\lambda^{\prime}\in\Lambda$ and $c\in\mathbb{Z}^{N}$ we have $c+\Delta_{\lambda}=\Delta_{\lambda^{\prime}}$. Then $\lambda-\lambda^{\prime}$ is a multiple of $(1,\ldots,1)$, that is, $\lambda$, $\lambda^{\prime}$ represent the same weight in $\Lambda^{\prime}$.

Proof.

Let the $(x_{ij})$, $(x^{\prime}_{ij})$ denote the coordinates of points in $\Delta_{\lambda}$, $\Delta_{\lambda^{\prime}}$ respectively. Also let $c=(c_{ij})$. The assumption that $c+\Delta_{\lambda}=\Delta_{\lambda^{\prime}}$ implies that for all $1\leq i\leq n-1$, $\lambda_{i}\leq x_{1i}\leq\lambda_{i+1}$ if and only if $\lambda^{\prime}_{i}\leq x_{1i}+c_{1i}\leq\lambda^{\prime}_{i+1}$. It follows that $\lambda^{\prime}_{i}=\lambda_{i}+c_{1i}$ and $\lambda^{\prime}_{i+1}=\lambda_{i+1}+c_{1i}$ which in turn implies that $c_{1i}=c_{1(i+1)}$. This finishes the proof. ∎

Recall that a virtual polytope is a formal difference of two polytopes. The set of virtual polytopes in $\mathbb{R}^{N}$ forms an infinite dimensional $\mathbb{R}$-vector space. For a fan $\Sigma$ in $\mathbb{R}^{N}$ let $\mathcal{P}(\Sigma)$ denote the subgroup of virtual lattice polytopes in $\mathbb{R}^{N}$ generated by polytopes whose normal fan is $\Sigma$. The group $\mathcal{P}(\Sigma)$ contains a copy of the additive group $\mathbb{Z}^{N}$ as the virtual lattice polytopes whose support function is linear on the whole $\mathbb{R}^{N}$.

Corollary 1.9**.**

(1) The map $\lambda\mapsto\Delta_{\lambda}$ gives a homomorphism $\phi:\Lambda=\Lambda(\textup{GL}(n,\mathbb{C}))\to\mathcal{P}(\Sigma_{\textup{GZ}})$. (2) The homomorphism $\phi$ induces an isomorphism $\bar{\phi}:\Lambda^{\prime}=\Lambda(\textup{SL}(n,\mathbb{C}))=\Lambda/\mathbb{Z}(1,\ldots,1)\to\mathcal{P}(\Sigma_{\textup{GZ}})/\mathbb{Z}^{N}$. (3) The quotient group $\Lambda^{\prime}$ is isomorphic to the Picard group of the toric variety $X_{\textup{GZ}}$ associated to the fan $\Sigma_{\textup{GZ}}$.

Proof.

The assertion (1) is an immediate corollary of Proposition 1.6. To prove (2) note that surjectivity of $\bar{\phi}$ follows from Proposition 1.3 and the injectivity of $\bar{\phi}$ is the content of Proposition 1.8. Finally, (3) follows from the well-known fact that for a fan $\Sigma$, the group $\operatorname{Pic}(X_{\Sigma})$ is isomorphic to the group $\textup{PL}(\Sigma,\mathbb{Z}^{N})$ of integer piecewise linear functions on $\Sigma$ modulo integer linear functions. This in turn can be identified with the quotient group $\mathcal{P}(\Sigma)/\mathbb{Z}^{N}$ (see [CLS11, Theorem 4.2.12]). ∎

2. Review of degrees of line bundles on toric and flag varieties

We recall that, for a projective variety $X$ of dimension $d$ embedded into a projective space $\mathbb{P}^{s}$, the degree of $X$ is defined to be:

[TABLE]

where the $H_{i}$ are generic hyperplanes in $\mathbb{P}^{s}$. Alternatively, let $[H]$ be the class of a hyperplane in $\text{Pic}(\mathbb{P}^{s})\cong\mathbb{Z}$ and let $[H^{\prime}]$ be the pullback of $[H]$ to $X$ via the embedding $X\hookrightarrow\mathbb{P}^{s}$, then $\text{deg}(X)=[H^{\prime}]^{d}$, the self-intersection number of the divisor class $[H^{\prime}]$.

If the embedding $X\hookrightarrow\mathbb{P}^{s}$ is given by the sections of a very ample line bundle $\mathcal{L}$, that is, $X\hookrightarrow\mathbb{P}(H^{0}(X,\mathcal{L})^{*})$, we will write $\deg(X,\mathcal{L})$ for $\deg(X)$. The asymptotic Riemann-Roch theorem, implies that

[TABLE]

If $\mathcal{L}$ is not very ample, we still define $\deg(X,\mathcal{L})$ as the self-intersection number of the divisor class of $\mathcal{L}$.

In the case $X=X_{\Sigma}$ is the toric variety of a fan $\Sigma$, we recall that all divisors are linearly equivalent to $T$-invariant divisors which in turn are generated by codimension $1$ orbit closures $D_{\rho}=\overline{O}_{\rho}$, $\rho\in\Sigma(1)$. Thus an arbitrary $T$-invariant divisor on $X_{\Sigma}$ can be written in the form $D=\sum_{\rho}a_{\rho}D_{\rho}.$ The associated line bundle will be $\mathcal{L}=\mathcal{O}(D)$, and the dimension of $H^{0}(X,\mathcal{L})$ is equal to the number of lattice points in the polytope $P_{D}=\{m\ |\ \langle m,v_{\rho}\rangle\geq-a_{\rho},\,\forall\rho\in\Sigma(1)\}$ where $v_{\rho}$ is the primitive vector along the ray $\rho$. One can also start with a lattice polytope $P$ normal to the fan of $X_{\Sigma}$. The support numbers $\{a_{\rho}\}_{\rho\in\Sigma(1)}$ of the polytope enable us to define a $T$-invariant divisor $D_{P}=\sum_{\rho\in\Sigma(1)}a_{\rho}D_{\rho}$ on $X_{\Sigma}$, and $P_{D_{P}}=P$. One shows that $D$ is ample that is, $kD$ defines an embedding into projective space for sufficiently large $k\in\mathbb{N}$. We have the following (which is a version of the well-known Bernstein-Kushnirenko-Khovanskii theorem):

Proposition 2.1**.**

Let $\mathcal{L}_{P}$ be the line bundle associated to the divisor $D_{P}$. Then:

[TABLE]

Proof.

By the asymptotic Riemann-Roch we have:

[TABLE]

∎

As we are interested in comparing $X_{\textup{GZ}}$ with the flag variety $\mathcal{F}\ell_{n}$, we also recall some facts about degrees of embeddings for $\mathcal{F}\ell_{n}$. Recall that to a weight $\lambda$ one associates a line bundle $\mathcal{L}_{\lambda}$ on $\mathcal{F}\ell_{n}$. This line bundle satisfies the property

[TABLE]

Similarly to the proof of Proposition 2.1, we can show the following (see for example [Kav11, Remark 2.4]).

Proposition 2.2**.**

For any dominant weight $\lambda$ we have:

[TABLE]

where $N=n(n-1)/2=\dim(\mathcal{F}\ell_{n})$.

Proof.

By the construction of the Gelfand-Zetlin polytope [GZ50], for every dominant $\lambda$, we have $\#(\Delta_{\lambda}\cap\mathbb{Z}^{N})=\dim(V_{\lambda})=\dim(V_{\lambda}^{*})$. On the other hand, by the Borel-Weil theorem, one knows that $H^{0}(\mathcal{F}\ell_{n},\mathcal{L}_{\lambda})\cong V_{\lambda}^{*}$. We note that for any $m>0$, $\mathcal{L}_{\lambda}^{\otimes m}=\mathcal{L}_{m\lambda}$ and $m\Delta_{\lambda}=\Delta_{m\lambda}$. Then the asymptotic Riemann-Roch theorem gives us:

[TABLE]

as required. ∎

Proposition 2.2 and Proposition 2.1 show that the map $\operatorname{Pic}(\mathcal{F}\ell_{n})\to\operatorname{Pic}(X_{\textup{GZ}})$, given by $\mathcal{L}_{\lambda}\mapsto\mathcal{L}_{\Delta_{\lambda}}$, preserves degree of line bundles. This observation is important in the proof of our main theorem (Theorem 5.1).

3. Review of intersection theory on toric and flag varieties

In this section we recall some basic facts about Chow rings and Chow cohomology rings of toric and flag varieties.

For an algebraic variety $X$ and $1\leq k\leq n=\dim(X)$, the $k$-th Chow group $A_{k}(X)$ is the group generated by algebraic $k$-cycles on $X$, that is, formal sums of irreducible $k$-dimensional subvarieties in $X$, modulo rational equivalence. Two $k$-cycles are equivalent if their difference is the divisor of a rational function on a $(k+1)$-dimensional subvariety, and the rational equivalence is the equivalence relation generated by this. The total Chow group of $X$ is $A_{*}(X)=\bigoplus_{k=0}^{n}A_{k}(X)$. When $X$ is smooth we let $A^{k}(X)=A_{n-k}(X)$ and $A^{*}(X)=\bigoplus_{k=0}^{n}A^{k}(X)$. In this case, the transverse intersection of subvarieties gives a well-defined multiplication on $A^{*}(X)$ making it into a graded algebra called the Chow ring of $X$ ([Ful13, Proposition 8.3]). More generally, for a commutative ring $R$, one can define the Chow groups $A_{k}(X,R)$ and the Chow ring $A^{*}(X,R)$ whenever $X$ is smooth.

In general, for a smooth variety $X$, the cohomology ring $H^{*}(X)$ and the Chow ring $A^{*}(X)$ are different. Nevertheless, for some nice varieties $X$ these algebras are naturally isomorphic ([Ful13, Example 19.1.11]).

Theorem 3.1**.**

Suppose $X$ is smooth and has a paving by affine cells, then $H^{*}(X)$ and $A^{*}(X)$ are naturally isomorphic.

The above theorem in particular applies to complete smooth toric varieties and the flag variety $\mathcal{F}\ell_{n}$.

When $X=X_{\Sigma}$ is a smooth complete toric variety, there is a nice description of the Chow ring $A^{*}(X_{\Sigma})$. In this case, for each $k$, the Chow group $A^{k}(X_{\Sigma})=A_{n-k}(X_{\Sigma})$ is generated by the orbit closures of codimension $k$. Although not needed in this paper, we state the following well-known result on description of the Chow ring of a smooth complete toric variety (see [Ful93, Section 5.2]).

Theorem 3.2**.**

Let $X_{\Sigma}$ be a smooth complete toric variety. Let $D_{1},\ldots,D_{r}$ be the codimnesion $1$ orbit closures corresponding to rays $\rho_{1},\ldots,\rho_{r}\in\Sigma(1)$. Then $A^{*}(X_{\Sigma})\cong H^{*}(X_{\Sigma})\cong\mathbb{Z}[D_{1},\ldots,D_{r}]/I$ where $I$ is the ideal generated by the following relations:

• (1)

$D_{i_{1}}\cdots D_{i_{k}}$* for all $\rho_{i_{1}},\ldots,\rho_{i_{k}}$ not contained in any cone of $\Sigma$ and,*

• (2)

$\sum_{i=1}^{r}\langle u,v_{\rho_{i}}\rangle D_{i}$* for all $u\in M$.*

There is also a nice description of the ring $A^{*}(\mathcal{F}\ell_{n})\cong H^{*}(\mathcal{F}\ell_{n})$ due to Borel. For each weight $\lambda$ let $c_{1}(\mathcal{L}_{\lambda})$ be the divisor class (Chern class) of the line bundle $\mathcal{L}_{\lambda}$ on $\mathcal{F}\ell_{n}$ (see [Bri05], in particular Remark 1.4.2 in there).

Theorem 3.3**.**

We have the following:

• (1)

The map $\lambda\mapsto c_{1}(\mathcal{L}_{\lambda})$ gives an isomorphism of $A^{1}(\mathcal{F}\ell_{n})=\operatorname{Pic}(\mathcal{F}\ell_{n})$ with the weight lattice $\Lambda^{\prime}=\Lambda(\textup{SL}(n,\mathbb{C}))=\Lambda/\mathbb{Z}(1,\ldots,1)$.

• (2)

$A^{*}(\mathcal{F}\ell_{n})$* is generated, as an algebra, by $c_{1}(\mathcal{L}_{\lambda})$, $\lambda\in\Lambda$.*

• (3)

$A^{*}(\mathcal{F}\ell_{n})\cong\textup{Sym}(\Lambda^{\prime})/I_{W}$* where $I_{W}$ is the ideal generated by non-constant $W$-invariants.*

In the proof of our main theorem (Theorem 5.1) we will need parts (1) and (2) in Theorem 3.3.

Remark 3.4**.**

Alternatively, $H^{*}(\mathcal{F}\ell_{n},\mathbb{Q})$ can be viewed as the polytope algebra of the Gelfand-Zetlin family (see [Kav11, Corollary 5.3]). There it is shown that

[TABLE]

where $I$ is the ideal of polynomials which, when viewed as differential operators, annihilate the volume polynomial of the Gelfand-Zetlin polytopes. This description of the Chow ring of the flag variety is it is closely related to the proof of Theorem 5.1 but is not directly used there.

We note that the toric variety $X_{\textup{GZ}}$ is not smooth except when $n=1,2$ and hence we need a more general notion of the Chow ring that applies to non-smooth varieties as well. For a (not necessarily smooth) variety $X$ in [FM81] Fulton and MacPherson construct a variant of the Chow ring called the operational Chow ring or Chow cohomology ring $A^{*}(X)=\bigoplus_{k=0}^{n}A^{k}(X)$. When $X$ is smooth it coincides with the usual Chow ring. When $X=X_{\Sigma}$ is a complete toric variety one has $A^{k}(X_{\Sigma})=\textup{Hom}(A_{k}(X_{\Sigma}),\mathbb{Z})$. Moreover, the ring $A^{*}(X_{\Sigma})$ can be described purely in terms of combinatorial data of Minkowski weights, which are certain integer valued functions on the fan $\Sigma$. In Section 7 we will use this combinatorial description for some computations in the Chow cohomology of the Gelfand-Zetlin toric variety for $n=3$. Section 6 reviews the Minkowski weights description of the Chow cohomology ring.

4. Some algebra lemmas

Let $A=\bigoplus_{i=0}^{n}A^{i}$ be a graded ring over a field ${\bf k}$ which is finite dimensional as a ${\bf k}$-vector space and $A^{0}\cong A^{n}\cong{\bf k}$. Following [HW17], we call the graded subalgebra of $A$ generated by $A^{1}$, the Lefschetz subalgebra of $A$. We recall that $A$ has Poincaré duality if the multiplication maps

[TABLE]

are non-degenerate for all $i$. Our goal is to compare $A^{*}(\mathcal{F}\ell_{n})\cong H^{*}(\mathcal{F}\ell_{n})$, which has Poincaré duality, with the algebra $A^{*}(X_{\textup{GZ}})$, which in general does not. We start by observing how to get a Poincaré duality algebra from a general graded algebra.

Lemma 4.1**.**

Let $A=\bigoplus_{i=0}^{n}A^{i}$ with $A^{0}\cong A^{n}\cong{\bf k}$. There exists a homogeneous ideal $I\subset A$ such that $A/I$ has Poincaré duality and is the smallest homogeneous ideal (with respect to inclusion) with this property.

Proof.

Consider the ideal $I$ generated by all the homogeneous elements $x\in A$ such that

[TABLE]

It is straightforward to check that $I$ has the required properties. ∎

We call the algebra $A/I$ in Lemma 4.1, the Poincaré duality quotient $\text{PD}(A)$ of $A$. We next recall a useful algebra fact (see [Kav11, Theorem 1.1] and [Eis95, Exercise 21.7]) which we will need later. It states that a Poincaré duality algebra is determined by its top power polynomial.

Theorem 4.2**.**

Let $A=\bigoplus_{i=0}^{n}A^{i}$ be a finite dimensional graded algebra over a field ${\bf k}$ which is generated by $A^{1}$, satisfies $A^{0}\cong{\bf k}\cong A^{n}$, and has Poincaré duality. Fix a basis $\{a_{1},\ldots,a_{r}\}$ for $A^{1}$, and consider the polynomial $P:{\bf k}^{r}\to{\bf k}$ defined by

[TABLE]

Then we have an isomorphism of graded algebras

[TABLE]

where $\partial_{i}=\frac{\partial}{\partial x_{i}}$, and $I$ is the ideal of polynomials in the operators $\partial_{1},\ldots,\partial_{r}$ which annihilate $P$. The isomorphism sends each $a_{i}$ to the image of $\partial_{i}$ in ${\bf k}[\partial_{1},\ldots,\partial_{r}]/I$.

A generalization of Theorem 8.1 for commutative algebras $A$ with Poincaré duality that are not necessarily generated by $A^{0}\cong{\bf k}$ and $A^{1}$ can be found in [EKK].

We now use Theorem 4.2 to to prove the following key lemma required in the proof of our main result (Theorem 5.1).

Lemma 4.3**.**

Suppose $A=\bigoplus_{i=0}^{n}A^{i}$ and $B=\bigoplus_{i=0}^{n}B^{i}$ are ${\bf k}$-algebras which are finite dimensional ${\bf k}$-vector spaces and have the following properties:

• (1)

$A^{0}\cong A^{n}\cong B^{0}\cong B^{n}\cong{\bf k}$.

• (2)

$A$* and $B$ are generated in degree one.*

• (3)

$A$* has Poincaré duality.*

• (4)

There exists a linear isomorphism $\varphi:A^{1}\to B^{1}$ such that for all $a_{1},\ldots,a_{n}\in A^{1}$ we have:

[TABLE]

using fixed isomorphisms $A^{n}\cong{\bf k}\cong B^{n}$.

Then $\varphi$ extends to give a ${\bf k}$-algebra isomorphism $\tilde{\varphi}$ between $A$ and the Poincaré duality quotient of $B$.

Proof.

We apply Theorem 4.2 to $A$ and to the Poincaré duality quotient $\text{PD}(B)$. Since $A$ already satisfies the conditions of Theorem 4.2 we know that $A\cong{\bf k}[\partial_{1},\ldots,\partial_{r}]/I$ where $r=\dim_{\bf k}(A^{1})$ and $I$ is the annihiliator of the top power polynomial $P$ described in Theorem 4.2. We need to show that $\text{PD}(B)$ also satisfies these conditions. First note that $B^{0}\cong{\bf k}\cong B^{n}$ so the multiplication $B^{0}\times B^{n}\to B^{n}\cong{\bf k}$ is already non-degenerate and thus the ideal $I$ in Lemma 4.1 contains neither $B^{0}$ nor $B^{n}$. This gives us $\text{PD}(B)^{0}\cong{\bf k}\cong\text{PD}(B)^{n}.$ Also, by construction $\text{PD}(B)$ has Poincaré duality. Finally, $\text{PD}(B)$ is generated in degree one since $B$ is generated in degree $1$. Now consider the map on degree one pieces:

[TABLE]

where $q$ is the quotient map. It suffices to show $\tilde{\varphi}:=q\circ\varphi:A^{1}\to\text{PD}(B)^{1}$ is an isomorphism. Since $\varphi$ is an isomorphism and $q$ is surjective, $\tilde{\varphi}$ is surjective and we only need to verify injectivity. Suppose for contradiction that some nonzero $a\in A^{1}$ has image $\tilde{\varphi}(a)=q(\varphi(a))=0$. Then $b=\varphi(a)$ is in the ideal in Lemma 4.1, so it is a linear combination of the $x_{i}$ satisfying $x_{i}\cdot B^{n-\deg(x_{i})}=0.$ Since $b\in B^{1}$, the $x_{i}$ must be in degree [math] or $1$. One knows that $B^{0}\cap I=\{0\}$, so we can only have $x_{i}\in B^{1}$. It follows that $b\cdot B^{n-1}=0$. But the assumption (4) then implies that $a\cdot A^{n-1}=0$ which contradicts that $A$ has Poincaré duality. Thus $\text{PD}(B)$ satisfies the conditions required for Theorem 4.2, and hence $\text{PD}(B)\cong{\bf k}[\partial_{1},\ldots,\partial_{r}]/I$. We have already seen that $A$ is isomorphic to this quotient algebra and thus $A\cong\text{PD}(B)$. ∎

5. Main theorem

We now state and prove our main theorem relating the cohomology ring of the flag variety $\mathcal{F}\ell_{n}$ and the Chow cohomology ring of the toric variety $X_{\textup{GZ}}$ associated to the GZ fan $\Sigma=\Sigma_{\textup{GZ}}$.

Theorem 5.1**.**

The cohomology ring $H^{*}(\mathcal{F}\ell_{n},\mathbb{Q})\cong A^{*}(\mathcal{F}\ell_{n},\mathbb{Q})$ is isomorphic to the Poincaré duality quotient of the Lefschetz subalgebra of $A^{*}(X_{\textup{GZ}},\mathbb{Q})$. For each dominant weight $\lambda$, the isomorphism sends the divisor class of the line bundle $\mathcal{L}_{\lambda}$ on $\mathcal{F}\ell_{n}$ to the image of the cohomology class in $X_{\textup{GZ}}$ associated to the GZ polytope $\Delta_{\lambda}$.

Proof.

We claim that there is an isomorphism of groups $A^{1}(\mathcal{F}\ell_{n})\cong A^{1}(X_{\textup{GZ}})$. One knows that $A^{1}(\mathcal{F}\ell_{n})=A_{N-1}(\mathcal{F}\ell_{n})=\operatorname{Pic}(\mathcal{F}\ell_{n})\cong\Lambda(\textup{SL}(n,\mathbb{C}))=\Lambda(\textup{GL}(n,\mathbb{C}))/\mathbb{Z}(1,\ldots,1)$. Also for a complete toric variety $X_{\Sigma}$, where $\Sigma$ is a complete fan in $\mathbb{R}^{N}$, the Chow cohomology group $A^{1}(X_{\Sigma})$ is naturally isomorphic to $\operatorname{Pic}(X_{\Sigma})$ (see [FS97, Corollary 3.4]). Now the claim follows from Corollary 1.9.

One knows that for an $N$-dimensional toric variety $X_{\Sigma}$, under the isomorphism $A^{1}(X_{\Sigma})\cong\operatorname{Pic}(X_{\Sigma})$ the top product of an element in $A^{1}(X_{\Sigma})\cong\operatorname{Pic}(X_{\Sigma})$ coincides with the self-intersection number of the corresponding divisor in $\operatorname{Pic}(X_{\Sigma})$. Applying this to the Gelfand-Zetlin toric variety $X_{\textup{GZ}}$, from Propositions 2.1 and 2.2, we now conclude that the isomorphism $\operatorname{Pic}(\mathcal{F}\ell_{n})=\operatorname{Pic}(X_{\textup{GZ}})$ respects the multiplication, i.e., it satisfies the assumption (4) in Lemma 4.3 (alternatively this can be deduced from [JY16, Theorem 4.3 and Corollary 4.5]). Applying Lemma 4.3 to $A=A^{*}(\mathcal{F}\ell_{n})$ and $B=$ the Lefschetz subalgebra of $A^{*}(X_{\textup{GZ}})$ finishes the proof.

∎

6. Minkowski weights

In this section we recall the description of the Chow cohomology ring of a toric variety in terms of Minkowski weights (see [FS97], see also [Kaz03]). We will use it in Section 7 to compute the Gelfand-Zetlin Chow cohomology ring for $n=3$. Let $\Sigma$ be a complete fan in $N$. Recall that $\Sigma(k)$ is the set of cones of dimension $k$ in $\Sigma$.

Definition 6.1**.**

A function $c:\Sigma(n-k)\to\mathbb{Z}$ is a Minkowski weight of codimension $k$ on $\Sigma$ if it satisfies the balancing condition for all $\tau\in\Sigma(n-k-1)$:

[TABLE]

Here $n_{\sigma,\tau}$ is a lattice point in $\sigma$ which generates the rank $1$ lattice $N_{\sigma}/N_{\tau}$, the quotient of the lattices spanned by $\sigma\cap N$ and $\tau\cap N$ respectively.

Let $MW^{k}$ denote the set of all Minkowski weights of codimension $k$. For two Minkowski weights $c\in MW^{p}$ and $\tilde{c}\in MW^{q}$, the product $c\cup\tilde{c}\in MW^{p+q}$ is defined by:

[TABLE]

where $m_{\sigma,\tau}^{\gamma}=[N:N_{\sigma}+N_{\tau}]$, and the sum is over all pairs of cones $(\sigma,\tau)$ which both contain $\gamma$ and $\sigma$ meets $\tau+v$ for fixed generic vector $v$ (see [FS97, Theorem 4.2]).

In [FS97] an isomorphism between the ring of Minkowski weights and the operational Chow ring of a complete toric variety $X_{\Sigma}$ is given. In fact it is shown that $MW^{k}\cong A^{k}(X_{\Sigma})$ (see [FS97, Theorem 3.1]). In particular:

[TABLE]

Example 6.2** (Hypersimplex).**

The following is an example of a fan where the ring $MW^{*}$ is not generated by $MW^{1}$ (see [FS97, Example 3.5] or [KP08, Example 4.2]). Consider the fan $\Sigma_{H}$ over the cube in $\mathbb{R}^{3}$ with vertices $(\pm 1,\pm 1,\pm 1)$. The rays in the fan $\Sigma_{H}$ are:

[TABLE]

One computes that $MW^{1}\cong\mathbb{Z}$ and $MW^{2}\cong\mathbb{Z}^{5}$. Thus $MW^{*}$ is not generated by $MW^{1}$.

7. Gelfand-Zetlin example, $n=3$

In this section we compute the Chow cohomology ring of $X_{\textup{GZ}}$ for $n=3$ using the Minkowski weights and show that while it is generated in degree $1$, it does not have Poincaré duality. We consider the GZ polytope of the weight $\lambda=(-1,0,1)$ for ease of computation. The polytope $\Delta_{\lambda}$ is defined by the following array of inequalities

[TABLE]

and has normal fan $\Sigma_{\textup{GZ}}$ as in Figure 2. We enumerate the rays as follows:

[TABLE]

Likewise, we let $\sigma_{ij}$ denote the 2-dimensional cone spanned by rays $\rho_{i}$ and $\rho_{j}$:

[TABLE]

Similarly, the collection of 3-dimensional cones are:

[TABLE]

We now compute $MW^{k}$, $k=0,\ldots,3$. A Minkowski weight in $MW^{3}$ is any map $\{0\}\to\mathbb{Z}$ and hence $MW^{3}\cong\mathbb{Z}$. A Minkowski weight $c\in MW^{2}$ is a function on rays $\rho_{i}$. Let $c(\rho_{i})=c_{i}$, then the single relation coming from the cone $\tau=0$ is given by $\sum_{i=1}^{6}c_{i}v_{\rho_{i}}=0$. From this we get the three relations:

[TABLE]

We see from this that any weight $c\in MW^{2}$ is determined by its values on three rays $c(\rho_{2})=a$, $c(\rho_{4})=b$ and $c(\rho_{6})=c$. Thus $MW^{2}\cong\mathbb{Z}^{3}$.

Next take $c\in MW^{1}$. It is a function on codimension $1$ cones $\sigma_{ij}$. Let $c(\sigma_{ij})=c_{ij}$. The relations among the $c_{ij}$ come from the rays. The relation for $\tau=\rho_{1}$ involves the cones $\sigma_{13},\sigma_{15}$ and $\sigma_{16}$. Let $n_{\sigma\tau}$ be the lattice point in $\sigma$ which generates the one-dimensional lattice $N_{\sigma}/N_{\tau}$. We compute:

[TABLE]

where all vectors are considered modulo $\rho_{1}=(1,0,0)$. The balancing condition then becomes

[TABLE]

which implies $c_{13}=c_{15}=c_{16}$. Similar computations for the other rays yield the following results:

[TABLE]

For later computations, we let:

[TABLE]

Finally, a weight $c\in MW^{0}$ is a function on top-dimensional cones subject to relations coming from $2$-dimensional cones. Each $2$-dimensional cone $\sigma_{ij}$ separates two top-dimensional cones, and the corresponding relation gives equality between the values of $c$ on each pair of top-dimensional cones. Hence $MW^{0}\cong\mathbb{Z}$ as the value of $c$ on each $3$-dimensional cone must be the same. In summary, we have the following:

[TABLE]

Before understanding the product structure on $MW^{*}$, it is already clear that this ring does not have Poincaré duality as the rank of $MW^{2}$ is greater than that of $MW^{1}$.

Recall from Section 6 that for weights $c\in MW^{p}$, $\tilde{c}\in MW^{q}$, their product is a function on cones of codimension $p+q$, and its value on a cone $\gamma\in\Sigma(3-p-q)$ is given by

[TABLE]

where the sum is over certain pairs $(\sigma,\tau)\in\Sigma(3-p)\times\Sigma(3-q)$ and $m_{\sigma\tau}^{\gamma}=[N:N_{\sigma}+N_{\tau}]$. We compute products of Minkowski weights in our example to determine whether $MW^{*}(X_{\textup{GZ}})$ is generated in degree $1$. Let $c,\tilde{c}\in MW^{1}(X_{\textup{GZ}})$ with:

[TABLE]

The Minkowski weight $c\cup\tilde{c}\in MW^{2}$ is evaluated on rays and from the arguments above it is enough to determine the value of this weight on the rays $\rho_{2}$, $\rho_{4}$ and $\rho_{5}$. Moreover, in Equation (6) for $(c\cup\tilde{c})(\rho_{2})$ the sum is over all pairs $(\sigma,\tau)\in\Sigma(2)\times\Sigma(2)$ where $\sigma$ and $\tau$ both contain $\rho_{2}$ and $\sigma$ meets $\tau+v$ for a generic fixed $v\in N$. The cones in $\Sigma(2)$ which contain $\rho_{2}$ are $\{\sigma_{23},\sigma_{24},\sigma_{25},\sigma_{26}\}$, so $\sigma,\tau$ will come from this collection. Since all these cones involve $\rho_{2}=(-1,0,0)$, we can sketch the relevant cones in the $yz$-plane where for example $\sigma_{23}$ can be viewed as $\rho_{3}=(1,0)$.

In Figure 3, we see the cones for $c$ in blue, and for $\tilde{c}$ in green using a shift of $v=(0.1,0.1,0.1).$ Then there are two pairs $(\sigma,\tau)$ which meet for this vector $v$, either $(\sigma,\tau)=(\sigma_{23},\sigma_{25})$ or $(\sigma,\tau)=(\sigma_{26},\sigma_{24})$. The last ingredient required to compute this product are the coefficients $m_{\sigma\tau}^{\rho_{2}}$ for the sum. In both cases, $N_{\sigma}+N_{\tau}=N$ so $m_{\sigma\tau}^{\rho_{2}}=1$. Thus we have

[TABLE]

Similar computations for $(c\cup\tilde{c})(\rho_{4})$ and $(c\cup\tilde{c})(\rho_{5})$ yield:

[TABLE]

Thus we see that the products $c\cup\tilde{c}$ in fact generate the entire $3$-dimensional space $MW^{2},$ and hence $MW^{*}$ for $\Sigma_{\textup{GZ}}$ is generated in degree $1$ for the case $n=3$.

Finally, the second author has written a Sage code which shows that for $n=4$, $5$, the ring $MW^{*}$ of $\Sigma_{\textup{GZ}}$ is not generated in degree $1$, and moreover its Lefschetz subalgebra does not have Poincaré duality. It can be found at https://github.com/evillella/minkowski. Also see the appendix in [Vil19].