Yang-Baxter algebras as convolution algebras: The Grassmannian case

TL;DR

This paper links quantum integrable models with Schubert calculus by explicitly constructing solutions to the Yang-Baxter equation using geometric methods related to Grassmannians, revealing new algebraic structures.

Contribution

It provides a geometric construction of Yang-Baxter solutions as convolution algebras from equivariant Schubert calculus, connecting integrable models with algebraic geometry.

Findings

Identified Yang-Baxter algebra as convolution algebra from Grassmannian Schubert calculus

Constructed quotients of current algebra rak{gl}_2[t] acting on tensor products

Connected the approach with the Cohomological Hall algebra (COHA) for the A_1-quiver

Abstract

We present a simple but explicit example of a recent development which connects quantum integrable models with Schubert calculus: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated Yang-Baxter algebras which play a central role in quantum integrable systems and exactly solvable lattice models in statistical physics. We consider the degenerate five-vertex limit of the asymmetric six-vertex model and identify its associated Yang-Baxter algebra as convolution algebra arising from the equivariant Schubert calculus of Grassmannians. We show how our method can be used to construct (Schur algebra type) quotients of the current algebra $\mathfrak{gl}_2[t]$ acting on the tensor product of copies of its evaluation representation $\mathbb{C}^2[t]$. Finally we connect it with the COHA for the $A_1$-quiver.

Equations

$${\mathbb{V}}_N=V^{\otimes N}\otimes \mathbf{P}=V^{\otimes N}[t_1,t_2,\dots ,t_N].$$

$$(V[t]{)}^{\otimes N}=V^{\otimes N}[t_1,t_2,\dots ,t_N]={\mathbb{V}}_N.$$

$$L(x,t)$$

$$M(x,t_1,\dots ,t_N)$$

$$v_{\lambda }=v_{{\lambda }_1}\otimes v_{{\lambda }_2}\otimes \cdots \otimes v_{{\lambda }_N},$$

$$M(x,t_1,\dots ,t_N)$$

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$$A\leftrightarrow (0,0),B\leftrightarrow (1,0),C\leftrightarrow (0,1),D\leftrightarrow (1,1).$$

$$L_{23}(x_2,t)L_{13}(x_1,t)R_{12}(x_1,x_2)$$

$$M_2(x_2,t_1,\dots ,t_N)M_1(x_1,t_1,\dots ,t_N)R_{12}(x_1,x_2)$$

$$\cdots$$

$$\left(\begin{array}{cccc}{\omega }_1& 0& 0& 0\\ 0& {\omega }_3& {\omega }_5& 0\\ 0& {\omega }_6& {\omega }_4& 0\\ 0& 0& 0& {\omega }_2\end{array}\right)$$

$$R(x,y):=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& x-y& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$$

$$L(x,t)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& x+t& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$$

$${\Delta }_1=\frac{{\omega }_1{\omega }_2+{\omega }_3{\omega }_4-{\omega }_5{\omega }_6}{2{\omega }_1{\omega }_3},\text{and}{\Delta }_2=\frac{{\omega }_1{\omega }_2}{{\omega }_1{\omega }_3}$$

$$({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4,{\omega }_5,{\omega }_6)$$

$$O(x)=\sum _{i\ge 0}{O}^{(i)}{x}^i$$

$$R_{12}(x_1,x_2)M_1(x_1,t)M_2(x_2,t)$$

$$A(x_1)A(x_2)=A(x_2)A(x_1),$$

$$B(x_1)A(x_2)=B(x_2)A(x_1),$$

$$C(x_1)C(x_2)=C(x_2)C(x_1),$$

$$B(x_2)D(x_1)=B(x_1)D(x_2),$$

$$B(x_1)C(x_2)=B(x_2)C(x_1),$$

$$A(x_2)B(x_1)-A(x_1)B(x_2)$$

$$C(x_2)A(x_1)-C(x_1)A(x_2)$$

$$A(x_1)D(x_2)-A(x_2)D(x_1)$$

$$D(x_2)B(x_1)-D(x_1)B(x_2)$$

$$C(x_1)D(x_2)-C(x_2)D(x_1)$$

$$C(x_2)B(x_1)-C(x_1)B(x_2)$$

$$v_{{\lambda }_3}$$

$$R_{12}(x_1,x_2)(v_0\otimes v_0\otimes A(x_1)B(x_2)\mathbf{v}+v_0\otimes v_1\otimes A(x_1)D(x_2)\mathbf{v}$$

$$(x_1-x_2)(v_0\otimes v_0\otimes B(x_2)A(x_1)\mathbf{v}+v_0\otimes v_1\otimes D(x_2)A(x_1)\mathbf{v}$$

$$A(x_2)B(x_1)-A(x_1)B(x_2)$$

$$C(x_1)D(x_2)-C(x_2)D(x_1)$$

$$C(x_2)B(x_1)-C(x_1)B(x_2)$$

$$A(x_1)D(x_2)-A(x_2)D(x_1)$$

Full text

1. 1 Diagrammatic from integrable systems
2. 2 Yang-Baxter algebras and Bethe vectors
3. 3 The $R$-matrix from the geometry of Grassmannians
4. 4 The geometric Yang-Baxter algebra via convolution in cohomology
5. 5 Equivariant quantum cohomology
6. 6 Connection with the current algebra $\mathfrak{gl}_{2}[t]$
7. 7 Connection to COHAs

Introduction

Quantum groups first arose in the physics literature, particularly in the work of the Faddeev school, from the so-called quantum inverse scattering method, [KBI93], [BIK85], [Fad85], which had been developed to construct and solve quantum integrable systems. In their original form, quantum groups are associative algebras whose defining relations are expressed in terms of a matrix of constants (depending on the integrable system under consideration) called a quantum R-matrix. It was realised independently by Drinfeld [Dri87] and Jimbo [Jim86] around 1985, that these algebras are Hopf algebras, which, in many cases, are deformations of universal enveloping algebras of Lie algebras. Thus, although many of the fundamental papers on quantum groups are written in the language of integrable systems, their properties are accessible by more conventional mathematical techniques, such as the theory of topological and algebraic groups and of Lie algebras.

The task of placing quantum groups in the setup of representations of quivers and geometric representation theory, was initiated by several researchers including Ringel [Rin90], Beilinson, Lusztig, MacPherson [BLM90] and Ginzburg [Gin91] just to name a few. The quantum groups studied by Drinfeld and Jimbo were hereby interpreted as the cohomology of certain complex algebraic varieties or as algebras of functions on certain varieties defined over finite fields. The associative algebra structure comes in both cases from a convolution product construction, in other words, these quantum groups were realised as subalgebras of some convolution algebra.

In the last years several groups of researchers suggested a further geometrisation of the theory around quantum R-matrices (for instance in the work of Nekrasov and Shatashvilli, [NS09], Braverman, Maulik and Okounkov, [BMO11], [MO12], [Oko15], Schiffmann and Vasserot [SV17]) leading into a slightly different direction. All these approaches provide a geometric construction of (an action of) $R$-matrices. Arising naturally as solutions of the Yang-Baxter equations, $R$-matrices are the major ingredient of the construction of a Yang-Baxter algebra and therefore it allows a geometric construction of Yang-Baxter algebras, often realised inside geometric constructions of quantum groups.

In the current paper we will present some new results in this direction, based on an explicit example. The starting point is the observation (Proposition 3.12 and Corollary 3.13) that the transition matrix between two natural bases in the equivariant cohomology of a Grassmannian is a solution to the Yang-Baxter equation. These two bases are formed by the classes of the attracting manifolds of two different one-dimensional torus actions (obtained by choosing two regular cocharacters of the natural $n$-dimensional torus, separated by a wall of the Weyl chamber). This is parallel to the approach of Maulik and Okounkov, [MO12], where the classical rational $R$-matrix serves as the transition matrix between different choices of stable envelope bases, a basis of the equivariant cohomology of the cotangent bundle to Grassmannians which naturally arises in the context of symplectic geometry and quiver varieties.

We instead work with the Grassmannians themselves and construct an action of the $R$-matrix (see Corollary 3.12) on equivariant cohomologies. Working with the base space instead of the cotangent bundle might be viewed as passing to an interesting degeneration of the existing constructions. Taking the $R$-matrix as an input for the so-called $\operatorname{RTT}=\operatorname{TTR}$-construction from [FRT90] we produce two Yang-Baxter algebras, (Definition 2.2 and Proposition 2.1), which we will study in this paper. These Yang-Baxter algebras are both a degeneration of the so-called asymmetric six vertex model studied in statistical physics, see (1.10), which allows us to connect it with the combinatorics and with Bethe vectors known in the theory of integrable systems, see also Remark 4.9.

Our first main result (Theorem 4.5) describes these Yang-Baxter algebras as an explicit subalgebra of a convolution algebra acting on the equivariant cohomology $H^{*}_{T}=\bigoplus_{k=0}^{N}H^{*}_{T}(\operatorname{Gr}(k,N))$ of the Grassmannian variety of all linear subspaces in a fixed vector space of dimension $N$. The two algebras are very similar, one involves twists with the Chern classes of the tautological bundle, the other Chern classes of the quotient bundle, a symmetry which was not obvious to us from the algebraic definition.

The (ordinary and equivariant) cohomology algebra of Grassmannians has been studied under the name Schubert calculus for more than a hundred years, see e.g. [KL72] for an overview. In particular, its (rational) cohomology comes with two distinct bases over $H^{*}(\operatorname{pt})$ (the $T$-equivariant cohomology of a point), namely the so-called Schubert basis arising as the classes of the Schubert varieties (which are closures of cells in a natural cell decomposition or attracting cells for the canonical torus action), and on the other hand the torus fixed point basis.

To make the connection with the theory of quantum integrable systems we identify the Schubert basis in $H^{*}_{T}$ with the standard or spin basis of the tensor product $V[t]^{\otimes N}$ for a two dimensional vector space $V=\mathbb{C}^{2}$. It turns out, see Corollary 3.16, that the torus fixed point basis then corresponds to the basis of Bethe vectors, see [Bax82] and references therein, in the context of quantum integrable systems. Our results might provide some new insights into Schubert calculus.

Working with the interplay of different bases of $V^{\otimes N}$ (as $\mathfrak{sl}_{2}$-module) related to the geometry of Grassmannians is not new and was used for instance frequently as the guiding principle for categorified representations of the quantum group $U_{q}(\mathfrak{sl}_{2})$, see e.g. [FK97], [Sav03], [FKS06]. The focus there is however on the realisation of integral bases (like the canonical bases of Lusztig and Kashiwara) and the combinatorics of Kazdhan-Lusztig polynomials. In these frameworks, $V^{\otimes N}$ is either identified with the direct sum of certain vector spaces of functions related to the Grassmannians or with the Grothendieck group of the direct sum of the categories of perverse sheaves on the Grassmannians. In contrast, the basis of Bethe vectors is not integral.

Moreover, in our framework, the current algebra $\mathfrak{gl}_{2}[t]$ appears (instead of $\mathfrak{sl}_{2}$) with its action on the tensor product $V[t]^{\otimes N}$ of its evaluation module for generic evaluation parameter, as for instance also in [RTSV11] (in the classical situation of $\mathfrak{gl}_{2}$ the situation would become trivial). We connect the universal enveloping algebra of the current Lie algebra $\mathfrak{gl}_{2}[t]$ in Theorem 6.12 with the two Yang-Baxter algebras. More precisely we realise the algebra of endomorphisms of $H^{*}_{T}$ (after some localisation) generated by these two algebras with a quotient of the universal enveloping algebra of $\mathfrak{gl}_{2}[t]$ localised with respect to a certain central subalgebra. This localisation is directly connected with the denominators appearing in the Bethe vectors and in the localisation theorem for equivariant cohomology. The connection is established via a Schur-Weyl duality type statement between this current Lie algebra and a subalgebra of the group algebra of the affine Weyl group, Corollary 6.10

This also illustrates that our setup can be viewed as an interesting degeneration of the situation described e.g. in [MO12] or [GRV92], where the Yangian and the affine Hecke algebra instead of the loop algebra and the group algebra of the affine Weyl group occur. The case of Yangians (and more general quantum groups) with a similar approach to ours was developed parallel and independently to our work in [SV17]. Our contribution should be seen as supplementary, giving an explicit example using classical geometry and elementary tools. We focus on the specific Grassmannian varieties, since we like to have explicit and elementary formulae. Our principal approach can however be generalised in the obvious way to arbitrary partial flag varieties of type $A$.

In the last section we finally connect our construction with cohomological Hall algebras (COHAs) in the sense of Kontsevich and Soibelman, [KS10]. We equip $H_{T}^{*}$ with an action of a (in fact the easiest possible) COHA and show in Theorem 7.3 that its image is contained in the endomorphism algebra generated by the two Yang-Baxter algebras.

Acknowledgement

We are grateful to C. De Concini, O. Foda, H. Franzen, L. Michalcea, R. Rimányi, N. Reshethikin, Y. Soibelman, V. Tarasov, A. Varchenko and P. Zinn-Justin for sharing ideas and knowledge generously, and T. Przezdziecki for comments on a draft version. We thank MPI and the HCM in Bonn where most of this research was done. The first author has been partially funded by the Russian Academic Excellence Project 5-100.

Conventions: For the whole paper we fix the ground field $\mathbb{C}$ of complex numbers and abbreviate $\otimes=\otimes_{\mathbb{C}}$, $\operatorname{Hom}=\operatorname{Hom}_{\mathbb{C}}$, $\operatorname{End}=\operatorname{End}_{\mathbb{C}}$ etc. For $A$, $B\in\operatorname{End}(V)$ we denote by $BA$ the composition $BA(v)=B(A(v))$ for $v\in V$. For a natural number $N$ set $[N]=\{1,2,\ldots,N\}$.

1. Diagrammatic from integrable systems

While the construction of the Yang-Baxter algebra from certain L-matrices is standard in the physics literature, we feel it is much less known to mathematicians. We therefore develop it here in detail adapted to our situation, and also refer to [HGK10] and [BNN14] for a similar setup.

For the whole paper we fix a natural number $N\geq 2$.

1.1. Lax matrices and monodromy matrices

In the following let $V=\mathbb{C}^{2}$ with a fixed basis $v_{0},v_{1}$. For any $\mathbb{C}$-algebra $S$ we denote by $\operatorname{Mat}_{2}(S)$ the vector space of $2\times 2$-matrices with entries in $S$. Any such matrix defines an endomorphism of $\mathbb{C}^{2}\otimes S$ in the $S$-basis $v_{0},v_{1}$. We often view elements in $\operatorname{Mat}_{2}(\operatorname{Mat}_{2}(S))$ as endomorphisms of $V\otimes V\otimes S=\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes S$ in the ordered basis $v_{0}\otimes v_{0}$, $v_{0}\otimes v_{1}$, $v_{1}\otimes v_{0}$, $v_{1}\otimes v_{1}$. For a finite set of variables $X$ denote by $\mathbb{C}[X]$ the polynomial ring in the variables $x\in X$ and let $W[X]:=W\otimes\mathbb{C}[X]$ for any vector space $W$. Set ${\bf P}=\mathbb{C}[t_{1},t_{2},\ldots,t_{N}]$ and $V_{N}=V^{\otimes N}$ and let

[TABLE]

Let $S_{N}$ be the symmetric group of order $N!$ generated by the simple transpositions $s_{i}=(i,i+1)$, $i\in[N-1]$ and $\mathbb{C}[S_{N}]$ its group algebra. It acts on ${\bf P}$ by algebra automorphisms $w(t_{i})=t_{w(i)}$. We denote by ${\bf P}^{S_{N}}$ the invariants, that is the subalgebra of ${\bf P}$ consisting of symmetric polynomials.

For $0\leq n\leq N$ let $\Lambda_{n}=\{(\lambda_{1},\ldots,\lambda_{N})\in\{0,1\}^{N}\mid\sum_{i=1}^{N}{\lambda_{i}}=n\}$ be the set of words of length $N$ in the letters [math], $1$ with $n$ $1$’s. Set $\Lambda=\bigcup_{n=0}^{N}\Lambda_{n}=\{0,1\}^{N}.$ In the following we identify the vector spaces

[TABLE]

by sending a vector $(w_{1}\otimes p_{1}(t))\otimes(w_{2}\otimes p_{2}(t))\otimes\cdots\otimes(w_{N}\otimes p_{N}(t))$ with $w_{i}\in V$ and $p_{i}(t)\in\mathbb{C}[t]$ to $(w_{1}\otimes w_{2}\otimes w_{N})\otimes(p_{1}(t_{1})p_{2}(t_{2})\cdots p_{N}(t_{N}))$.

Definition 1.1**.**

We pick a matrix $L(x,t)\in\operatorname{Mat}_{2}(\operatorname{Mat}_{2}(\mathbb{C}[x,t]))$, called Lax matrix, that is

[TABLE]

with entries in $\operatorname{Mat}_{2}(\mathbb{C}[x,t])$. It defines with our choice of basis a $\mathbb{C}[x,t]$-linear endomorphism of $V\otimes V[x,t]$.

Consider now the tensor product $V[x]\otimes V[t]^{\otimes N}$. We call $V[x]$ the [math]th factor and then number the remaining tensor factors from $1$ to $N$.

Definition 1.2**.**

The Monodromy matrix is the endomorphism

[TABLE]

of $V[x]\otimes V[t]^{\otimes N}$, where $L_{ij}$ is the endomorphism acting as $L(x,t)$ on the tensor factors $i$ and $j$ and as the identity on the remaining factors.

In the standard ${\bf P}[x]$-basis, or spin basis, defined as

[TABLE]

$M(x,t_{1},t_{2},\ldots,t_{N})$ can be viewed as a block $2\times 2$-matrix with blocks, $A=A(x,t_{1},t_{2},\ldots,t_{N})$ etc. of size $2^{N}\times 2^{N}$, i.e.

[TABLE]

and the blocks are matrices of ${\bf P}[x]$-linear operators acting on $V[x]\otimes\mathbb{V}_{N}$.

To calculate the matrix entries of (1.4) and algebraic relations between them we use some diagrammatic calculus which is a common tool in the theory of integrable systems. For this we first identify the standard basis vectors (1.3) with their $\{0,1\}$-words $(\lambda_{1},\lambda_{2},\dots,\lambda_{N})$ of length $N$. A crossing is a diagram of the form as shown on the left of (1.5) built out of an ordered pair of directed line intervals intersecting in their midpoints. The $4$ line segments obtained by removing the intersection point are called edges. The directions assigned to the lines split the edges into inputs and outputs. The order of the lines puts an order on the inputs and on the outputs. In the picture, the edge on the left is the first input, the one on the top the second input, the edge on the right is the first output and the one on the bottom the second output. We will often omit drawing the arrows and the ordering.

A labelled crossing is a crossing, where all the four edges are labelled by an element from $\{0,1\}$ which we also display by drawing the edges solid if they are labelled with $1$ and dotted if they are labelled with [math]. For example, the following two pictures represent the same labelled crossing with input pair (solid, dotted) and output pair (dotted, solid).

[TABLE]

The 16 possible labelled crossings are displayed in the matrix below.

[TABLE]

A (labelled) lattice diagram is a diagram obtained by composing finitely many labelled crossings vertically and horizontally by gluing an output edge with an input edge of the same colour, see e.g. (1.7), together with a total ordering on the lines. If not otherwise specified, the ordering is always taking the left inputs from bottom to top followed by the top inputs from left to right, see (1.7).

Definition 1.3**.**

A Lax matrix $L\in\operatorname{Mat}_{2}(\operatorname{Mat}_{2}(\mathbb{C}[x,t]))$ assigns to a labelled crossing a (Boltzmann) weight, namely the $(i,j)$-entry of the matrix $L$ if the crossing is as the $(i,j)$-entry in (1.6). The weight of a labelled lattice diagram is then the product of all weights of the crossings, but evaluated at $x=x_{a},t=t_{b}$ if the $a$-th row and $b$-th column cross. If there is only one row (resp. one column) then we usually keep the variable $x$ (resp. t) instead of evaluating at $x_{1}$ (resp. $t_{1}$).

Example 1.4**.**

In case of $L=L(x,t)$ respectively $L=L^{\prime}(x,t)$ from (1.11) below, we have the following: the weight of the purely black crossing equals $1$ respectively $x-t$. The weight of the lattice diagram in the middle of (1.8) equals $x+t_{2}$ respectively $x-t_{1}$, whereas for the one on the right it is $1$ and $(x-t_{1})(x-t_{2})$ respectively.

1.2. Calculating matrix entries

It is not hard to check now that the matrix entries in (1.4) can be calculated as follows: consider a $1$-row lattice of length $N$, that is a labelled lattice diagram obtained by putting $N$ labelled crossings next to each other, e.g. for $N=3$,

[TABLE]

The top sequence of labels defines a $\{0,1\}$-word $\lambda_{in}$, and thus a vector $v_{\lambda_{in}}\in V_{N}$, called the input vector; likewise, the bottom sequence of labels defines the output vector $v_{\lambda_{out}}\in V_{N}$. The two (left and right) outer horizontal edges specify one of the operators $A$, $B$, $C$, $D$ from (1.4) via the assignments

[TABLE]

The corresponding Boltzmann weight is then a polynomial in ${\bf P}[x]$, and for the Lax matrices (1.11) of total degree at most $N$. From the definitions we deduce:

Proposition 1.5**.**

Let $O\in\{A,B,C,D\}$ and $v_{\lambda_{in}},v_{\lambda_{out}}\in V^{\otimes N}$ standard basis vectors. Then the matrix coefficient $(O(v_{\lambda_{in}}),v_{\lambda_{out}})$ is equal to the sum of the weights of all $1$-row lattice diagrams of length $N$ whose external edge labelling is fixed and given by the triple ($O$, $v_{\lambda_{in}}$, $v_{\lambda_{out}}$).

Example 1.6**.**

Let us calculate the coefficient $c$ of $Cv_{(0,1,1)}$ at $v_{(0,1,0)}$. The labelling of the external edges is as displayed in (i) which for $L(x,t)$ or $L^{\prime}(x,t)$ as in (1.11) can be extended to a unique lattice diagram with non-zero weight as in (ii), and thus $x+t_{2}$ respectively $x-t_{1}$ is its weight.

[TABLE]

In comparison, the coefficient of $Cv_{(0,0,1)}$ at $v_{(0,0,0)}$ equals the weight of (iii), see Example 1.4.

1.3. Yang-Baxter equation

Definition 1.7**.**

A pair $(R(x,y),L(x,t))$ from $\operatorname{Mat}_{2}(\operatorname{Mat}_{2}(\mathbb{C}[x,t]))$ is a solution of the Yang-Baxter equation if the following holds

[TABLE]

as endomorphisms of $V^{\otimes 3}[x_{1},x_{2},t]$. The subindices indicate the factors on which the respective operator acts non-trivially.

By incorporating $R$, the diagrammatics from above can easily be adopted to express (1.9) graphically. To write the identity (1.9) as a system of equalities for the matrix entries of the above $4\times 4$ matrices with blocks of size $2\times 2$, attach to $R(x,y)$ a St. Andrew’s cross ${\times}$ with first input at the bottom left and the second input at the top left, and consider the following two diagrams.

[TABLE]

Here the three lines correspond to the three tensor factors involved in (1.9), with the inputs numbered as indicated. Reading the diagrams from left to right and top to bottom, the crossings correspond precisely to the factors involved in the compositions (1.9) (with the inputs always on the left and at the top and the outputs on the bottom and to the right). By a labelling of such a diagram we mean, as before, a diagram with labels $\{0,1\}$ attached to all the edges, and the weight of such a diagram is again defined as the product of the weights attached to the vertices (except that now the new type of crossings get weights from the matrix $R$).

The following is a consequence of the definitions:

Proposition 1.8**.**

A pair $(R(x,y),L(x,t))$ from $\operatorname{Mat}_{2}(\operatorname{Mat}_{2}(\mathbb{C}[x,t]))$ gives a solution of the Yang-Baxter equation (1.9) if and only if the following holds: for each common labelling of the external edges of the diagrams (LHS) and (RHS) the respective sums of weights (for all possible complete labellings) agree.

Proposition 1.9**.**

If $(R(x,y),L(x,t))$ is a solution to the Yang-Baxter equation, then the identity

[TABLE]

holds in $\operatorname{End}_{P[x_{1},x_{2}]}(V\otimes V\otimes\mathbb{V}_{N}[x_{1},x_{2}])$ for any $N\geq 1$, where $M_{i}$ acts on the $i$th factor $V$ and on $\mathbb{V}_{N}[x_{1},x_{2}]$ as in (1.2).

Proof.

This follows from Proposition 1.8 by the usual rail road principle, which means diagrammatically that we move the crossing representing $R$ to the right

[TABLE]

by applying iteratively the Yang-Baxter equation from Proposition 1.8. ∎

1.4. Six and five vertex models

Usually one restricts to $R$-matrices which preserve the weight spaces if we view $V$ as the vector representation of $\mathfrak{sl}_{2}(\mathbb{C})$. Matrices of this form

[TABLE]

appear for instance in the asymmetric six vertex model, [Bax82], in the physics literature.

Definition 1.10**.**

The two examples of $R$-matrices and Lax matrices, which will be treated here explicitly, are the following

[TABLE]

[TABLE]

The Lax operators (1.11) are so-called 5-vertex degenerations, see also Remark 4.9, of the asymmetric 6-vertex model which is used to model ferroelectrics in external electromagnetic fields [Bax82] and are representatives of the two classes studied e.g. in [PR06] in the theory of quantum integrable systems, [HWKK96], going back to [Wu68]. The pair $(R,L)$ is called in [Kor14] (observing the switch in notation) the osculating walkers model, and the pair $(R^{\prime},L^{\prime})$ the vicious walkers model.

Remark 1.11**.**

It is known that a (generic) pair $(R(x,y),L(x,t))$ of matrices, each of the form (1.10), is a solution of (1.9), if the Boltzmann weights $(\omega_{1},\omega_{2},\omega_{3},\omega_{4},\omega_{5},\omega_{6})$ for each matrix yield constant values for the following two ratios, that is

[TABLE]

are well-defined complex numbers for either of the two matrices. This criterion is originally due to Baxter [Bax82], but can be also found in [BBF11, Theorem 2]. Note that this criterion fails for the (degenerate) vicious walker model.

Remark 1.12**.**

There are two types of transformations one can perform on the weights of a solution of (1.9) which produces a new solution, namely

[TABLE]

This follows easily diagrammatically, since the first exchanges [math] and $1$ labels on the vertical and the horizontal edges of each vertex configuration, whereas the second exchanges [math] and $1$ on the vertical edge and swaps the entries on the horizontal edge.

Proposition 1.13**.**

The pairs $(R(x,y),L(x,t))$ and $(R^{\prime}(x,y),L^{\prime}(x,t))$ are both solutions to the Yang-Baxter equation (1.9).

Proof.

For $(R^{\prime},L^{\prime})$ this follows directly from Remark 1.11, as we find $\Delta_{2}=0$ in these cases. Applying the transformation (1.13) to $(R^{\prime},L^{\prime})$ and replacing $t$ by $-t$ we obtain the other pair $(R,L)$. By Remark 1.12 it solves the Yang-Baxter equation as well. ∎

2. Yang-Baxter algebras and Bethe vectors

From now on we work with the solutions $(R,L)$ and $(R^{\prime},L^{\prime})$ from (1.11) to the Yang-Baxter equation. We define now the main object of our studies.

2.1. Yang-Baxter algebras

Let $O=O(x,t_{1},\ldots,t_{N})\in\{A,B,C,D\}$. Expand it as

[TABLE]

in the variable $x$, with coefficients in $\operatorname{End}_{{\bf P}}(\mathbb{V}_{N})=\operatorname{End}_{\bf P}(V^{\otimes N}[t_{1},t_{2},\ldots,t_{N}])$.

Proposition 2.1**.**

In case of $(R,L)$, the identity from Proposition 1.9,

[TABLE]

is equivalent to the following sixteen relations

[TABLE]

A similar reformulation of the Yang-Baxter equations can be given for $(R^{\prime},L^{\prime})$.

Proof.

We will illustrate how to obtain the claim, but omit the straight-forward calculations. It suffices to assume $N=3$. We have to consider all possible inputs in (2.1) and compare their outputs. Let us consider the input $(0,1,\lambda_{1},\lambda_{2},\lambda_{3})$, with $\lambda_{i}\in\{0,1\}$.

[TABLE]

Then we obtain with ${\bf v}=v_{\lambda}$ for the diagram on the right hand side of (2.3)

[TABLE]

The output of the left hand side of (2.3) is

[TABLE]

Comparing the coefficients of the basis vectors we obtain

[TABLE]

These arguments applied to all possible inputs give the asserted 16 relations. ∎

Note that in fact each of the 16 equations is a system of equations for the operators $\eqref{ABCD}$. Following [FRT90] we use them to define abstract algebras:

Definition 2.2**.**

The Yang-Baxter algebra $\operatorname{YB}$ (or $\operatorname{YB}^{\prime}$) is the ${{\bf P}}$-algebra generated by $\hat{A}^{(i)}$, $\hat{B}^{(i)}$, $\hat{C}^{(i)}$, $\hat{D}^{(i)}$ (respectively by $\hat{A^{\prime}}^{(i)}$, $\hat{B^{\prime}}^{(i)}$, $\hat{C^{\prime}}^{(i)}$, $\hat{D^{\prime}}^{(i)}$), $i\in\mathbb{Z}_{\geq 0}$ modulo the relations (see Proposition 2.1) given by the Yang-Baxter equation (2.2) for $(R,L)$ respectively $(R^{\prime},L^{\prime})$ from (1.11).

These algebras act on $\mathbb{V}_{N}$, by sending the generators to the endomorphisms

[TABLE]

from (2.1) for all $i\geq 0$. In the following we will often omit the lower index $N$.

Note that the space $V^{\otimes{N}}$ has a natural decomposition $V^{\otimes{N}}=\bigoplus_{i=0}^{N}V_{n}$ into subspaces where $V_{n}$ is spanned by all standard basis vectors $v_{\lambda}$ from (1.3) such that $\lambda\in\Lambda_{n}$. Similarly, we have $\mathbb{V}_{N}=\bigoplus_{n=0}^{N}\mathbb{V}_{N,n}$, the corresponding decomposition into free ${\bf P}$-modules, which we call weight spaces. The action of $\operatorname{YB}$ is compatible with the weight space decomposition in the following sense.

Lemma 2.3**.**

Consider the monodromy matrix for $(R,L)$ and $(R^{\prime},L^{\prime})$ as in (1.11). For every $i\geq 0$ and $0\leq n\leq N$ it holds

[TABLE]

Proof.

We only argue for $(R,L)$, the other case is similar. For $N=1$ the matrix $M_{1}(x,t_{1})$ is the Lax matrix, hence the generators of the algebra $\operatorname{YB}_{1}$ are the operators given by the coefficients of the expansion in $x$ of the $2\times 2$ blocks of $L(x,t)$, explicitly

[TABLE]

which obviously have the asserted property. The definition of the monodromy matrix (1.2) implies immediately the claim for any $N$. ∎

2.2. Symmetric group action and ${\bf H}$-action

The symmetric group $S_{N}$ acts on $\mathbb{V}_{N}=V\otimes{\bf P}$, such that $w\in S_{N}$ sends $v_{\lambda}\otimes f$ to $w(v_{\lambda})\otimes w(f)=v_{\lambda^{\prime}}\otimes w(f)$ with $\lambda^{\prime}_{j}=\lambda_{w^{-1}(j)}$, that is, $w$ simultaneously commutes the indices of the basis vector and the variables of the polynomials.111Some readers might prefer to define ${\bf s}_{i}=\sigma R(t_{i},t_{i+1})P_{i,i+1}$, where $P_{i,i+1}$ is the ${\bf P}$-linear flip operator on $V^{\otimes N}$ and $\sigma$ is the permutation of the variables $t_{i}$ and $t_{i+1}$. Since our $R$-matrices (1.11) satisfy $s_{1}R(t_{2},t_{1})s_{1}R(t_{2},t_{1})=\operatorname{id}_{\mathbb{V}_{2}}$, the operators

[TABLE]

for $i\in[N-1]$ give rise to an action of the symmetric group $S_{N}$ on $\mathbb{V}_{N}$. This action is not ${\bf P}$-linear, but part of an action of the semidirect product algebra ${\bf H}$ (or skew group ring) which is defined as follows, where we from now on write ${}^{w}\!f$ instead of $w(f)$ when $w\in S_{N}$ is applied to $f\in{\bf P}$.

Definition 2.4**.**

The algebra ${\bf H}$ is the vector space ${\bf H}=\mathbb{C}[S_{N}]\otimes{\bf P}$ with the multiplication such that $\mathbb{C}[S_{N}]$ and ${{\bf P}}$ are subalgebras and $fw={}^{w^{-1}}\!\!fw$ for $w\in S_{N},f\in{{\bf P}}$.

Explicitly, $f\in{\bf P}$ acts by multiplication with $f$, and the action of ${\bf s}_{i}$ is given as follows.

Proposition 2.5**.**

On the standard ${\bf P}$-basis vectors $v_{\lambda}$ we have

[TABLE]

which then extends to the desired action of ${\bf H}$ by $s_{i}(fv_{\lambda})={}^{s_{i}}\!f\;{\bf s}_{i}(v_{\lambda})$.

Remark 2.6**.**

Note that the result of $\operatorname{id}-{\bf s}_{i}$ is divisible by $t_{i}-t_{i+1}$, hence the quotient is well-defined. It is the well-known Demazure operator $\Delta_{i}$, [Dem74], a standard tool, see e.g. [Kum02], [KT03], in cohomology theory, see also Section 3.3.

Our Yang-Baxter algebras have the important property, pointed out already in [GoKo17], that they commute with this $S_{N}$-action. More precisely the following holds.

Proposition 2.7**.**

Let $N\geq 2$ and $i\in[N-1]$. Then there is an equality

[TABLE]

of endomorphisms of $V[x]\otimes\mathbb{V}_{N}$, with monodromy matrix $M$ for $L$ or $L^{\prime}$ from (1.11).

Proof.

The statement is local and involves only three tensor factors, namely the leftmost factor at position [math] and the factors $i$ and $i+1$ in $\mathbb{V}_{N}$. We may assume $i=1$. First consider the Yang-Baxter equation

[TABLE]

Substituting $x_{2}=-t_{1}$, $t=t_{2}$ we obtain

[TABLE]

using the special shape of $L$ and $R$, e.g. $L(-x,y)=R(y,x)$. Thus

[TABLE]

For the pair $(R^{\prime},L^{\prime})$ the following relation can be verified by a direct calculation

[TABLE]

Since the symmetric group action is defined as ${\bf s}_{i}=s_{i}R^{\prime}_{i,i+1}(t_{i+1},t_{i})$, the proof that it commutes with the monodromy matrix $M^{\prime}$ is the same as above. ∎

2.3. Bethe vectors

The Yang-Baxter algebra contains several interesting commutative subalgebras. In the theory of quantum integrable systems there is a procedure, called the (algebraic) Bethe Ansatz or the quantum inverse scattering method, [KBI93], [BIK85], for finding a common eigenbasis. In general this is a rather sophisticated procedure, see e.g. [Tak85] for a general introduction, but for the purposes of this paper it suffices to consider the simplest version of it.

The relation $A(x_{1})A(x_{2})=A(x_{2})A(x_{1})$ in the Yang-Baxter algebra means that the coefficients of the operator $A(x)$ commute, and hence can be diagonalised simultaneously. The key relation for doing this is

[TABLE]

Lemma 2.8**.**

Let $y=(x,y_{1},\ldots,y_{k})$ be a $k+1$-tuple of pairwise distinct commuting variables. Then we have the following formal identities:

[TABLE]

where the superscript $i$ indicates the $i$th factor.

Proof.

We do induction using (2.7) with starting point (2.7) itself. Suppose

[TABLE]

holds. Then $A(x)C(y_{k})\cdots C(y_{1})$ equals

[TABLE]

Hence the proof will be finished if we show that

[TABLE]

Since the operators $C(x)$ and the $C(y_{i})$’s pairwise commute, the numerators on both sides coincide. Therefore the claim is reduced to the equality

[TABLE]

which can easily be verified. ∎

The Bethe Ansatz is a method of finding eigenvectors for the $A^{(i)}_{N}$, $i\geq 0$ from (2.1) of the form

[TABLE]

depending on tuples $\eta=(\eta_{1},\ldots\eta_{k})\in{\bf P}^{k}$. The parameters are solution to a system of algebraic equations called the Bethe Ansatz equations which are hard to compute in general. However in our case the solutions are easy to find using Lemma 2.8 and the identity

[TABLE]

Note that permuting the components of $\eta$ in (2.9) provides the same vector, since the operators $C(y_{i})$ pairwise commute.

Proposition 2.9**.**

For $\eta=(-t_{i_{1}},\ldots,-t_{i_{k}})$ where $I_{0}=\{i_{j}\mid 1\leq j\leq k\}\subseteq[N]$ with distinct elements,

[TABLE]

is a simultaneous eigenvector for the $A^{(i)}_{N}$’s with eigenvalues given by

[TABLE]

The $\mathbf{b}(\eta)$ for ${i_{1}}<i_{2}<\cdots<{i_{k}}$ form a basis of $\mathbb{V}_{N,n}$, the basis of Bethe vectors.

Proof.

Indeed setting $y_{j}=-t_{i_{j}}$ for $j=1,\cdots,k$ in the statement of Lemma 2.8 and applying the operators to the vector $v_{(1,1,\dots,1)}$, we see that on the RHS only the first summand survives due to (2.10). Then (2.12) follows. ∎

Example 2.10**.**

Let $k=1$. Then a typical diagram representing a matrix element of the operator $C(x)$ acting on the vector $v_{(1,\ldots,1)}$ looks as follows

[TABLE]

and more precisely, we obtain the formula

[TABLE]

and via evaluation at $x=-t_{i_{1}}$ for $1\leq i_{1}\leq N$, the corresponding Bethe vector

[TABLE]

expressed in the standard basis. In particular $\mathbf{b}(0,1,\ldots,1)=v_{(0,1,\ldots,1)}$.

Lemma 2.11**.**

For $\lambda$ of the form $(0,\ldots,0,1,\ldots,1)$ the corresponding Bethe vector equals the standard basis vector, $\mathbf{b}_{(0,\ldots,0,1,\ldots,1)}=v_{(0,\ldots,0,1,\ldots,1)}$.

Proof.

The only diagram contributing to the Bethe vector (2.11) indexed by $(0,\ldots,0,1,\ldots,1)$ is

[TABLE]

Since the product of the weights in this diagram is equal to $1$, we are done. ∎

We need the localisation of ${\bf P}$ and $\mathbb{V}_{N}$ with respect to the multiplicative set $\operatorname{loc}$ given by products of the form $(t_{a}-t_{b})$, where $a,b\in[N]$,

[TABLE]

to make sense of this normalisation (giving rise to normed Bethe vectors for an appropriate form).

Definition 2.12**.**

Given ${\bf\lambda}\in\Lambda$ let $I_{0}=\{i\mid\lambda_{i}=0\}\subset[N]$. Then the associated (normalised) Bethe vector $\mathbf{b}_{\lambda}\in\mathbb{V}_{N}^{\operatorname{loc}}$ is the vector $\mathbf{b}(\eta)$ from (2.11) normalised as

[TABLE]

The (normalised) Bethe basis of $\mathbb{V}_{N}^{\operatorname{loc}}$ is the ${\bf P}^{\operatorname{loc}}$-basis given by $\mathbf{b}_{\lambda}$, $\lambda\in\Lambda$.

The $S_{N}$-action (2.5) extends to $\mathbb{V}_{N}^{\operatorname{loc}}$ and just permutes the Bethe basis vectors:

Proposition 2.13**.**

The action of $S_{N}$ from (2.5) satisfies for $j\in[N-1]$

[TABLE]

Proof.

The $S_{N}$-action from (2.5) commutes with the Yang-Baxter operators by Proposition 2.7 and ${\bf s}_{j}(v_{(1,\ldots,1)})=v_{(1,\ldots,1)}$ by the definition of the $R$-matrix. Therefore, acting by ${\bf s}_{j}$ on the vector $C(-t_{i_{k}})\ldots C(-t_{i_{1}})v_{(1,\ldots,1)}$ amounts to applying the permutation $s_{j}$ to the $-t_{i_{j}}$’s and the normalisation factor. ∎

Remark 2.14**.**

For the pair $(R^{\prime},L^{\prime})$, the arguments are completely analogous. Instead of (2.7) one would consider the equation $B^{\prime}(x_{1})A^{\prime}(x_{2})-B^{\prime}(x_{2})A^{\prime}(x_{1})=(x_{2}-x_{1})A^{\prime}(x_{1})B^{\prime}(x_{2})$. Then the vectors $\mathbf{b}^{\prime}(\eta)=B^{\prime}(t_{i_{1}})\cdots B^{\prime}(t_{i_{n}})v_{(0,0,\dots,0)}\in\mathbb{V}_{N}$ defined similar to (2.11) satisfy the equality $A^{\prime}(x)\mathbf{b}^{\prime}(\eta)=\displaystyle{\prod_{j\notin\{i_{1},\cdots,i_{n}\}}(x-t_{j})}\mathbf{b}(\eta)$. In fact, $\mathbf{b}^{\prime}(\eta^{\prime})=\mathbf{b}(\eta)$, where $\eta^{\prime}$ is complementary to $\eta$.

Remark 2.15**.**

The readers who are surprised about the fact that only the operators $A(x)$, and not the $D(x)$, occur in the construction of the Yang-Baxter algebra are referred to Section 5.

3. The $R$-matrix from the geometry of Grassmannians

We first recall some facts from the Schubert calculus for Grassmannians, see e.g. [Bri05], [Ful97] for more details. Then we will interpret $\mathbb{V}_{N}$ geometrically as $\bigoplus_{n=0}^{N}H_{T}^{*}({\operatorname{X}}_{n})$ identifying the standard basis with an equivariant Schubert basis. After some localisation, the normalised Bethe basis makes sense. It will correspond under the localised isomorphism to the easiest possible basis in geometry

[TABLE]

namely the fixed point basis for the $T$-action, as we show in Corollary 3.16.

3.1. Basics

We consider now $\mathbb{C}^{N}$, fix its standard basis $e_{1},\ldots,e_{N}$ and let $G=\operatorname{GL}_{N}(\mathbb{C})$. For $0\leq n\leq N$ we denote by ${\operatorname{X}}_{n}=\operatorname{Gr}(n,N)$ the corresponding Grassmannian variety of $n$-dimensional subspaces in $\mathbb{C}^{N}$. It is the projective variety $\operatorname{Gr}(n,N)=G/P$ where $P=P_{n,k}$ is the standard parabolic subgroup consisting of the block upper triangular invertible matrices with diagonal blocks of sizes $n$ and and $k:=N-n$ containing the standard Borel subgroup $B$ of all upper triangular matrices. Let $T\subset G$ be the torus of diagonal matrices. It acts on ${\operatorname{X}}_{n}$ because it acts obviously on $\mathbb{C}^{N}$, respectively by left multiplication on $G/P$. The set ${\operatorname{X}}_{n}^{T}$ of fixed points of this action is finite. Explicitly, there is a bijection $\Lambda_{n}\stackrel{{\scriptstyle 1:1}}{{\leftrightarrow}}{\operatorname{X}}_{n}^{T}$, $\lambda\mapsto\mathbb{C}^{\lambda}$, sending $\lambda\in\Lambda_{n}$ to the coordinate plane $\mathbb{C}^{\lambda}$ spanned by all $e_{i}$ with $\lambda_{i}=1$. Fix the standard flag

[TABLE]

where $F_{i}$ is spanned by $e_{j}$ for $j\in[i]$. For $\lambda\in\Lambda_{n}$ let $C_{\lambda}$ be the Schubert cell in ${\operatorname{X}}_{n}$,

[TABLE]

and let $\Omega_{\lambda}$ be its Schubert variety, i.e.  the closure of the cell $\operatorname{C}_{\lambda}\subset{\operatorname{X}}_{n}$. Note that $\operatorname{C}_{\lambda}$ contains exactly one $T$-fixed point, namely $\mathbb{C}^{\lambda}$. Define the inversion set of $\lambda\in\Lambda_{n}$ as

[TABLE]

and let $\ell(\lambda)=|\operatorname{inv}(\lambda)|$, e.g. $\ell((0,1,1))=0$, $\ell((1,0,1))=1$ and $\ell(1,1,0)=2$.

The Schubert class $\operatorname{S}_{\lambda}\in H^{*}({\operatorname{X}}_{n})=H^{*}({\operatorname{X}}_{n},\mathbb{C})$ is the Poincare dual in cohomology of the cycle in homology represented by the Schubert variety $\Omega_{\lambda}$. In particular, the degree of $\operatorname{S}_{\lambda}$ is $2\ell(\lambda)$. These classes are well known to form a basis (even over $\mathbb{Z}$) of $H^{*}({\operatorname{X}}_{n})$. Since the cells $C_{\lambda}$ are $T$-invariant (in fact $B$-orbits) they define also $T$-equivariant cohomology classes which we denote by abuse of notation also by $\operatorname{S}_{\lambda}$. They form a $H_{T}^{*}(\operatorname{pt})$-basis for the $T$-equivariant cohomology $H_{T}^{*}({\operatorname{X}}_{n})$. Later we will need a twisted version of this equivariant Schubert basis which we introduce now.

3.2. Twisted Schubert classes

Let $w\in S_{N}$. Acting with $w$ on the standard flag $\operatorname{F}^{e}$ produces a new flag $\operatorname{F}^{w}$, where $F^{w}_{i}$ is spanned by $e_{w(j)}$, $j\leq i$. Note that $\operatorname{F}^{e}$ corresponds to the neutral element $e\in S_{N}$. Mimicking the construction of $C_{\lambda}$ for the flag $\operatorname{F}^{w}$ instead of $\operatorname{F}^{e}$, we obtain another cell decomposition with cells

[TABLE]

Note that $\operatorname{C}^{w}_{\lambda}$ contains again exactly one fixed point, namely $\mathbb{C}^{w(\lambda)}$. The closure of $\operatorname{C}^{w}_{\lambda}$ is denoted $\Omega^{w}_{\lambda}$ and called the $w$-twisted Schubert variety.

Example 3.1**.**

Let us consider $\operatorname{Gr}(1,2)=\mathbb{P}^{1}$. The $T$-fixed points are represented by coordinate lines $p_{1}=\mathbb{C}e_{1}$ and $p_{2}=\mathbb{C}e_{2}$ corresponding to the $\{0,1\}$-words $\lambda=(1,0)$ and $\lambda=(0,1)$ respectively. For the standard flag $\operatorname{F}^{e}$ the Schubert cells are the single point, $C_{(1,0)}=\{p_{1}\}$, and the ‘big’cell, $C_{(0,1)}$, containing the fixed point $p_{2}$. For the flag $\operatorname{F}^{s_{1}}$ the cell $C^{s_{1}}_{(1,0)}$ is now the ‘big’ cell containing the fixed point $p_{1}$, and $C^{s_{1}}_{(0,1)}$ is the single point $p_{2}$.

The Schubert cells are the $B$-orbits in ${\operatorname{X}}_{n}$, namely $\operatorname{C}_{\lambda}=B\mathbb{C}^{\lambda}$. Moreover, $V\in\operatorname{C}^{w}_{\lambda}$ if and only if $\operatorname{dim}\left((w^{-1}(V)\cap F_{i})/(w^{-1}(V)\cap F^{w}_{i-1})\right)=\lambda_{w(i)}=(w^{-1}(\lambda))_{i}$ for all $i$, that is $w^{-1}(V)\in B\mathbb{C}^{w^{-1}(\lambda)}$ or equivalently $V\in wB\mathbb{C}^{w^{-1}(\lambda)}=wBw^{-1}w\mathbb{C}^{w^{-1}(\lambda)}$. Hence twisting with $w$ means taking the orbits for the conjugated Borel subgroup $B^{w}=wBw^{-1}$ in ${\operatorname{X}}_{n}$, in formulae

[TABLE]

Another convenient way, is to work with attracting cells for a $\mathbb{C}^{*}$-action. For this pick integers $a_{1},a_{2},\ldots,a_{N}$ and consider the corresponding integral cocharacter $\tau^{\bf a}:\mathbb{C}^{*}\rightarrow T,t\mapsto t^{\bf a}$, where $t^{\bf a}$ is the matrix with diagonal entries $t^{a_{1}},t^{a_{2}},\ldots,t^{a_{n}}$. Then there is the following alternative description:

Proposition 3.2**.**

Assume $a_{1}>a_{2}>\ldots>a_{N}$.

1. 1.)

Then the cell $\operatorname{C}_{\lambda}$ is the attracting manifold to the fixed point $p=\mathbb{C}^{\lambda}$, i.e.  $\operatorname{C}_{\lambda}=\{x\in{\operatorname{X}}_{n}\mid\lim_{t\rightarrow 0}\tau^{\bf a}(t).x=p\}.$ 2. 2.)

The cell $\operatorname{C}^{w}_{\lambda}$ is the attracting manifold to the fixed point $p=\mathbb{C}^{\lambda}$ for the cocharacter $t^{w^{-1}(\bf a)}$ corresponding to $a_{w^{-1}(1)},\ldots,a_{w^{-1}(N)}$.

Proof.

The first statement is clear and for the second note then that $w^{-1}(x)\in B\mathbb{C}^{w^{-1}(\lambda)}$ if and only if $\lim_{t\rightarrow 0}\tau^{\bf a}(t).w^{-1}(x)=\mathbb{C}^{w^{-1}(\lambda)}$ and so the statement follows from $w\tau^{\bf a}(t)w^{-1}=\tau^{w^{-1}(\bf a)}(t)$. ∎

Remark 3.3**.**

Note that if we view ${\bf a}$ as an element of the integral weight lattice for $\mathfrak{gl}_{N}$, then the attached attracting cell is constant on Weyl chambers.

3.3. Equivariant twisted Schubert classes

The cells $\operatorname{C}^{w}_{\lambda}$, for fixed $w$ and varying $\lambda$, define also equivariant cohomology classes $S^{w}_{\lambda}$ which, as we will explain now, form a $H_{T}^{*}(pt)$-basis, and $S^{e}_{\lambda}=S_{\lambda}$ will give the (untwisted) equivariant Schubert classes. For more details on equivariant cohomology we refer to [GKM98], [Lib07] and for our special case to [KT03].

To describe the $T$-equivariant cohomology of $H_{T}^{*}({\operatorname{X}}_{n})=H_{T}^{*}({\operatorname{X}}_{n},\mathbb{C})$ of ${\operatorname{X}}_{n}$ we will use the $T$-equivariant map ${\operatorname{X}}_{n}^{T}\rightarrow{\operatorname{X}}_{n}$ which is known to induce a monomorphism $\operatorname{res}:H_{T}^{*}({\operatorname{X}}_{n})\rightarrow H_{T}^{*}({\operatorname{X}}_{n}^{T})$ often called the restriction to the fixed points.

Let $\operatorname{Func}_{n}=\operatorname{Maps}(\Lambda_{n},{\bf P})$ be the set of functions on $\Lambda_{n}$ with values in ${\bf P}$. We identify ${\bf P}$ with $H^{*}_{T}(\operatorname{pt})$, and $\operatorname{Func}_{n}$ with $H^{*}_{T}({\operatorname{X}}_{n}^{T})$, and view $H^{*}_{T}({\operatorname{X}}_{n})\subset\operatorname{Func}_{n}$ via $\operatorname{res}$. Then, by [GKM98], see also [Lib07, Theorem 98], the subset $H_{T}^{*}({\operatorname{X}}_{n})\subset\operatorname{Func}_{n}(n)$ can be identified with the set of GKM-classes.

Definition 3.4**.**

An element $\alpha\in\operatorname{Func}_{n}$ is called a GKM class or just a class if it satisfies the following Goresky-Kottwitz-MacPherson (GKM) conditions:

[TABLE]

for a transposition $s_{ij}\in S_{N}$. For a class $\alpha$ define its support as $\operatorname{Supp}(\alpha)=\{\lambda\mid\alpha({\lambda})\not=0\}\subset\Lambda_{n}$.

We now give a combinatorial description of the twisted equivariant Schubert classes, generalising the (ordinary) equivariant Schubert classes from [KT03].

Lemma 3.5**.**

Consider the action of $S_{N}$ on $\operatorname{Func}_{n}$ in the usual way, namely

[TABLE]

for $w\in S_{N}$, $\alpha\in\operatorname{Func}_{n}$. Then the following holds.

1. 1.)

If $\alpha$ is a class, then so is $w\cdot\alpha$ for any $w\in S_{N}$. 2. 2.)

$\mu\in\operatorname{Supp}(\alpha)$* if and only if $w(\mu)\in\operatorname{Supp}(w\cdot\alpha)$.* 3. 3.)

$H_{T}^{*}({\operatorname{X}}_{n})\subset\operatorname{Func}_{n}$* is a subring and a ${\bf P}$-submodule.* 4. 4.)

The Demazure operators $\Delta_{i}:f\mapsto\frac{f-{}^{s_{i}}\!f}{t_{i}-t_{i+1}}$, $i\in[N-1]$, acting on $\operatorname{Func}_{n}$, preserve $H_{T}^{*}({\operatorname{X}}_{n})$.

Proof.

Except of the last part this follows directly from the definitions. For the last part we have to verify the GKM-conditions for $\Delta_{a}(\alpha)$ for any class $\alpha$. We have $\Delta_{a}(\alpha)(\lambda)-\Delta_{a}(\alpha)(s_{a}\lambda)=0$ if $\alpha$ is a class, hence the condition (3.3) holds for $\{i,j\}=\{a,a+1\}$. Otherwise $t_{i}-t_{j}$ and $t_{a}-t_{a+1}$ are coprime in ${\bf P}$. Since the difference $\alpha-s_{a}(\alpha)$ is a class, $\Delta_{a}(\alpha)$ is a class as well. ∎

Given $w\in S_{N}$, define a partial ordering ${\geq_{w}}$ on $\Lambda_{n}$ as follows.

Definition 3.6**.**

For $\lambda,\mu\in\Lambda_{n}$ let

[TABLE]

In particular $\geq_{\text{id}}\,=\,\geq$. We denote by $\Lambda_{\geq_{w}\lambda}$ all elements $\nu\in\Lambda_{n}$ such that $\nu\geq_{w}\lambda$.

Since $\mathbb{C}^{\mu}\in\Omega_{\lambda}^{w}$ if and only if $\mu\geq_{w}\lambda$, we are interested in the following classes.

Definition 3.7**.**

Let $w\in S_{N}$ and $\lambda\in\Lambda_{n}$. We say a class $\alpha\in\operatorname{Func}_{n}$ has $w$-support above $\lambda$ if $\lambda\in\operatorname{Supp}(\alpha)$ and $\operatorname{Supp}(\alpha)\subseteq\Lambda_{\geq_{w}\lambda}$.

The GKM-conditions imply that if $\alpha$ has $w$-support above $\lambda$, then $\alpha(\lambda)$ is divisible by $\prod_{(i,j)\in inv_{w(\lambda)}}(t_{i}-t_{j})$. We consider such classes of minimal total degree in the variables of ${\bf P}$.

Lemma 3.8**.**

Let $w\in S_{N}$, $\lambda\in\Lambda_{n}$. There exists a unique class $S^{w}_{\lambda}\in H^{*}_{T}({\operatorname{X}}_{n})$ satisfying

• Stab1:

$S^{w}_{\lambda}$* has $w$-support above $\lambda$,*

• Stab2:

$S^{\omega}_{\lambda}(\lambda)=\prod_{(i,j)\in\operatorname{inv}_{w}(\lambda)}(t_{j}-t_{i})$, where

[TABLE]

• Stab3:

$\operatorname{deg}(S^{w}_{\lambda}(\mu))=\ell(w^{-1}\lambda)$* for all $\mu\in\Lambda_{n}$ such that $\mu\geq_{w}\lambda$.*

This class is called the $w$-twisted Schubert class (or $w$-stable class) corresponding to $\lambda$.

Proof.

Note that $\ell(w^{-1}\lambda)$ equals the cardinality of the set

[TABLE]

Then the proof is exactly as in [KT03, Lemma 1], but with the order there replaced by $\geq_{w}$. ∎

Remark 3.9**.**

For $w=e$ we obtain the ordinary equivariant Schubert classes. In particular, $\Delta_{a}S^{e}_{\lambda}=S^{e}_{s_{a}(\lambda)}$ if $\ell(\lambda)>\ell(s_{a}\lambda)$ and $\Delta_{a}S^{e}_{\lambda}=0$ otherwise, see [KT03, Lemma 6].

Remark 3.10**.**

Alternatively one could define $S^{w}_{\lambda}(\mu)={}^{w}(S^{e}_{w^{-1}(\lambda)}(w^{-1}(\mu))$. Indeed, the characterizing properties Stab1 and Stab3 follows directly from the support condition for the untwisted Schubert classes. For the second one it suffices to show $w(\operatorname{inv}_{e}\omega^{-1}(\lambda))=\operatorname{inv}_{\omega}\lambda.$ To see this note that $(i,j)\in w(\operatorname{inv}_{e}\omega^{-1}(\lambda))$ if and only if $(w^{-1}(i),w^{-1}(j))\in\operatorname{inv}_{e}w^{-1}(\lambda)$. This is equivalent to $w^{-1}(i)<\omega^{-1}(j)$ with $w^{-1}(\lambda)_{w^{-1}(i)}>w^{-1}(\lambda_{w^{-1}(j)}$ or to $w^{-1}(i)<\omega^{-1}(j)$ with $\lambda_{i}>\lambda_{j}.$ But the latter means by definition that $(i,j)\in\operatorname{inv}_{w}(\lambda)$.

Example 3.11**.**

Assume the setup of Example 3.1. Then

[TABLE]

where the first line is always for $\mu=(10)$ and the second for $\mu=(01)$.

3.4. Wall-crossing

We now express, for fixed $w\in S_{N}$, the $s_{i}w$-twisted equivariant Schubert basis in terms of the $w$-twisted one for any simple transposition $s_{i}$. This generalises and gives a geometric interpretation of the formula [MS99, Proposition 3.2 with $l=1$] for factorial Schur functions.

Proposition 3.12**.**

Let $w\in S_{N}$ and $s=s_{i}=(i,i+1)$ then

[TABLE]

In particular, the $\{S^{w}_{\lambda}\}_{\lambda\in\Lambda_{n}}$ form, for fixed $w\in S_{N}$, a basis of $H^{*}_{T}({\operatorname{X}}_{n})$.

Proof.

We call the function on the right hand side $\alpha$ and rewrite $\alpha(\mu)$ using Remark 3.10. Abbreviating $\nu=w^{-1}(\lambda)$ we obtain

[TABLE]

In the first case we have $\ell(s\nu)>\ell(\nu)$ hence using Remark 3.9 we obtain $\alpha({\mu})={}^{w}((s\cdot S^{e}_{s\nu})(w^{-1}\mu))={}^{ws}(S^{e}_{s\nu}(sw^{-1}\mu))=S^{ws}_{\lambda}(\mu)$ as claimed. In the second case there are two possibilities $\nu_{i+1}=\nu_{i}$, or $\nu_{i+1}<\nu_{i}$ which means in either case $\Delta_{i}(S^{e}_{s\nu})=0$ by Remark 3.9. Hence $S^{e}_{s\nu}=s\cdot S^{e}_{s\nu}$ and therefore $w(S^{e}_{s\nu}(w^{-1}\mu))=ws(S^{e}_{s\nu}(sw^{-1}\mu))=S^{ws}_{\lambda}({\mu})$. This finishes the proof. ∎

Identify $H^{*}_{T}({\operatorname{X}}_{n})$ with $\mathbb{V}_{N,n}$ from Section 2 by sending $S^{e}_{\lambda}$ to $v_{\lambda}$ and extending ${\bf P}$-linearly. Then Proposition 3.12 can be formulated in terms of the $R$-matrix $R$ from (1.11) as follows.

Corollary 3.13**.**

The base change from $\{S^{ws_{i}}_{\lambda}\}_{\lambda\in\Lambda_{n}}$ to $\{S^{w}_{\lambda}\}_{\lambda\in\Lambda_{n}}$ is given by $R_{a,b}(t_{b},t_{a})$ acting on the $a$th and $b$th tensor factor of $\mathbb{V}_{N}$, where $a=w(i)$ and $b=w(i+1)$.

Remark 3.14**.**

Corollary 3.13 can be viewed as a construction of the $R$-matrix in the spirit of the geometric construction of the $R$-matrices of the Yangians in [MO12]. In light of Remark 3.3, Proposition  3.13 can be viewed as the equivariant behaviour of attracting manifolds when we cross a wall passing from one Weyl chamber to a neighboured one. In the non-equivariant version this wall-crossing collapses to a (boring) symmetric group action (permuting the factors of $V^{\otimes N}$).

By the equivariant localisation theorem, $H_{T}^{*}({\operatorname{X}}_{n})\subset\operatorname{Func}_{n}(n)$ becomes an isomorphism after localisation at $\operatorname{loc}$ with fixed point-basis (over ${\bf P}^{\operatorname{loc}}$) $\mathbbm{1}_{p}\in\operatorname{Func}_{n}(n)$, $p\in{\operatorname{X}}_{n}^{T}$, given by $\mathbbm{1}_{p}(p^{\prime})=\delta_{p,p^{\prime}}$.

3.5. Geometric Bethe basis

To connect the algebraic setup from the previous sections to the geometric one, we have to make (a common) twist, which means we consider instead of the ordinary Schubert basis the twisted Schubert basis $\mathbb{B}_{n}$ of $H^{*}_{T}(\operatorname{X})$ corresponding to the longest permutation

[TABLE]

By Stab 1 to Stab 3 the twisted Schubert class $S^{\omega_{0}}_{({0}^{k},{1}^{n})}\in\mathbb{B}_{n}$ takes the value

[TABLE]

on the fixed point $p_{n}\in{\operatorname{X}}_{n}^{T}$ corresponding to the unique maximal element $\zeta=({0}^{k},{1}^{n})=(\underbrace{0,\ldots,0}_{\text{k}},\underbrace{1,\ldots,1}_{\text{n}})\in\Lambda_{n}$ and is [math] on all other fixed points. The action (3.4) of the symmetric group in this basis is easy to calculate:

Lemma 3.15**.**

For any $i\in[N-1]$ and $\lambda\in\Lambda_{n}$ we have

[TABLE]

Proof.

Note that $s_{i}\omega_{0}=\omega_{0}s_{N-i}$ and $\omega^{-1}_{0}=\omega_{0}$ holds. Thus by Remark 3.10 ${s}_{i}\cdot S^{\omega_{0}}_{\lambda}={s}_{i}\cdot\omega_{0}\cdot S_{\omega_{0}(\lambda)}=(s_{i}\omega_{0})\cdot S_{\omega_{0}\lambda}=(\omega_{0}s_{N-i+1})\cdot S_{\omega_{0}(\lambda)}=\omega_{0}\cdot s_{N-i+1}\cdot S_{\omega_{0}(\lambda)}$. On the other hand, expressing (3.4) via the last part of Lemma 3.5, gives

[TABLE]

Thus we obtain

[TABLE]

This proves the claim ∎

Comparing it with (2.6) we obtain the following result: consider the localisations $\mathbb{V}_{N}^{\operatorname{loc}}$ from (2.13) and $H_{T}^{*}({\operatorname{X}}_{n})^{\operatorname{loc}}=\operatorname{Func}_{n}(n)^{\operatorname{loc}}$ as ${\bf P}^{\operatorname{loc}}$-modules.

Corollary 3.16**.**

Sending the standard basis element $v_{\lambda}$ to the twisted Schubert basis element $S^{\omega_{0}}_{\lambda}$ defines an isomorphism of ${\bf P}^{\operatorname{loc}}$-modules

[TABLE]

Under this isomorphism $\Psi$, the normalised Bethe vectors are sent to the fixed point basis vectors, in formulae $\Psi(\mathbf{b}_{\lambda})=\mathbbm{1}_{\operatorname{w}_{\lambda}}$, where $\operatorname{w}_{\lambda}$ is the torus fixed point corresponding to $\lambda$.

Proof.

The first claim follows directly from the definitions. By (2.6) and Lemma 3.15 the isomorphism $\Psi$ intertwines the two $S_{N}$-actions. But then the statements follow from Lemma 2.11 as the starting point together with Proposition 2.13, since (3.4) becomes the obvious permutation action of $S_{N}$ on torus fixed points. ∎

4. The geometric Yang-Baxter algebra via convolution in cohomology

Having established (via Corollary 3.13) a homology interpretation of the $R$-matrix $R(x,t)$, we will provide a similar interpretation of the corresponding Yang-Baxter algebra $\operatorname{YB}_{N}$.

4.1. Some combinatorics of torus fixed points in partial flag varieties

In order to state the main theorem we will need not only the Grassmannians ${\operatorname{X}}_{n}$ but also their generalisation, the partial flag varieties. For any positive integer $m$ denote by $\mathcal{P}_{m}(N)$ the set of compositions of $N$ with $m$ parts (possibly zero), e.g.  $\mathcal{P}_{2}(2)=\{(2,0),(1,1),(0,2)\}$. For $\mathbf{d}\in\mathcal{P}_{m}(N)$ let ${\operatorname{X}}_{\bf d}$ be the corresponding partial flag variety, i.e.  the projective variety given by all flags

[TABLE]

such that $\operatorname{dim}(F_{i+1}/F_{i})=d_{i}$. That is ${\operatorname{X}}_{\bf d}=G/P$ where $P$ is the parabolic subgroup given by all matrices $(a_{i,j})_{i,j}$ with $a_{i,j}=0$ if $j<\mu_{r}<i$ for some $r$, and $T$ still acts on it. There are $m$ natural $T$-equivariant vector bundles $M_{1},\ldots,M_{m}$ over ${\operatorname{X}}_{\bf d}$ of rank $d_{1},\ldots,d_{m}$ respectively; the fibre of $M_{i}$ at $\operatorname{F}$ is the vector space $F_{i+1}/F_{i}$. These bundles, as well as the tangent bundle $\operatorname{T}({\operatorname{X}}_{\bf d})$, are $T$-equivariant and thus have equivariant Chern classes which we will denote by $\mathfrak{c}_{i}(M)$, $\mathfrak{c}_{i}(\operatorname{T}({\operatorname{X}}_{\bf d}))$.

We recall some well-known facts we will need later. The main reference for us is [And10]. The set of $T$-fixed points ${\operatorname{X}}_{\bf d}^{T}\subset{\operatorname{X}}_{\bf d}$ is finite. To describe them explicitly let $\operatorname{F}^{e}_{\mathbf{d}}$ be the standard flag where $F_{i}$ is spanned by $\{e_{1},\ldots,e_{\mu_{i}}\}$, where $\mu_{i}=\sum_{j=1}^{i}d_{i}$. Clearly, $\operatorname{F}^{e}_{\mathbf{d}}$ is fixed under the action of $T$. The symmetric group $S_{N}$ acts on $\mathbb{C}^{N}$ by permuting the elements of the standard basis, so it acts on ${\operatorname{X}}_{\bf d}$. For any $w\in S_{N}$, the partial flag $\operatorname{F}^{w}_{\mathbf{d}}=w(\operatorname{F}^{e}_{\mathbf{d}})$ belongs to ${\operatorname{X}}_{\bf d}^{T}$, and in this way we get all fixed points. The stabiliser of $\operatorname{F}^{e}_{\mathbf{d}}$ coincides with the Young subgroup $S_{\mathbf{d}}=S_{d_{1}}\times S_{d_{2}}\times\ldots\times S_{d_{m}}$, so the assignment $w\mapsto\operatorname{F}^{w}_{\mathbf{d}}$ induces a bijection between $S_{N}/S_{\mathbf{d}}$ and ${\operatorname{X}}_{\bf d}^{T}$. In other words, a fixed point can be described as a partitioning $\operatorname{w}$ of the set $[N]$ into $m$ disjoint subsets $\operatorname{w}_{i}$ of cardinalities $d_{i}$ such that $F_{i}/F_{i-1}$ is spanned by the images of the elements $e_{l}$, $l\in\operatorname{w}_{i}$, see [Ful97] for more details.

Definition 4.1**.**

As a special case, a fixed point for ${\operatorname{X}}_{n}$ is determined by $\operatorname{w}_{1}$ (exactly the positions of $1$’s in the corresponding $\{0,1\}$-word), and then of $\operatorname{X}_{n,1,{n-k-1}}$ by a tuple $\operatorname{w}_{z}=(\operatorname{w}_{1},z)$ for some $z\in[N],l\notin\operatorname{w}_{1}$, and in $\operatorname{X}_{n+1}$ then by the union $\operatorname{w}_{1}\cup\{z\}$ which we abbreviate as $\operatorname{w}_{1}\cup z$. For $S\subset[N]$ let $\overline{S}$ be the complement in $[N]$.

For instance if $N=4$, then $X_{2}$ has six fixed points $\langle e_{i},e_{j}\rangle$, $1\leq i<j\leq 4$ corresponding to the partitioning $\operatorname{w}=\{i,j\}\cup\overline{\{i,j\}}$ and determined by $\operatorname{w}_{1}=\{i,j\}$. A fixed point in $X_{2,1,1}$ is then given by $\langle e_{i},e_{j}\rangle\subset\langle e_{i},e_{j},e_{z}\rangle$ for some distinct $i,j,k\in\{1,2,3,4\}$ and so determined by the tuple $(\{i,j\},z)$.

4.2. The geometric Yang-Baxter algebras

For $\operatorname{w}\in{\operatorname{X}}_{\bf d}^{T}$ let $\operatorname{inc}_{\operatorname{w}}:\{\operatorname{pt}\}\hookrightarrow{\operatorname{X}}_{\bf d}$, $\operatorname{pt}\mapsto\operatorname{w}$ be the $T$-equivariant inclusion. We have the following standard fact, see e.g. [And10, Example 2.1].

Lemma 4.2**.**

Let $\operatorname{w}=(\operatorname{w}_{1},\ldots,\operatorname{w}_{m})\in{\operatorname{X}}_{\bf d}^{T}$. Then the Euler class $\operatorname{eu}_{\operatorname{w}}$ of the fixed point $\operatorname{w}$ is

[TABLE]

where $\operatorname{top}$ is twice the dimension of $X_{\mathbf{d}}$ and $\operatorname{inc}_{\operatorname{w}}^{*}(\mathfrak{c}_{j}(M_{i}))=\sigma_{j}(t_{j_{1}},\ldots,t_{j_{d_{i}}}),$ where $\operatorname{w}_{i}=\{j_{1},\ldots,j_{d_{i}}\}$ and $\sigma_{j}$ denotes the $j$th elementary symmetric function.

For $m=1$, the partial flag variety is a point and for $m\leq 2$ we obtain the Grassmannian varieties ${\operatorname{X}}_{n}$. The inclusion $\operatorname{incl}:\operatorname{X}^{T}\hookrightarrow\operatorname{X}$ induces a monomorphism $\operatorname{incl}^{*}$ on equivariant cohomology which is exactly $\operatorname{res}$. Moreover $M_{1}$ is the tautological bundle $\mathcal{T}_{n}$, and $M_{2}$ the quotient bundle $\mathcal{Q}_{n}$. For the three step flag variety $\operatorname{X}_{(n,1,N-n-1)}$ we have a diagram

[TABLE]

where $\pi_{1}$ and ${\pi_{2}}$ are the obvious proper projection maps. It defines convolution operators acting on $H^{*}_{T}(\operatorname{X})$, where $\operatorname{X}=\bigcup_{n=0}^{N}{\operatorname{X}}_{n}$ is the disjoint union of all Grassmannians. The convolution algebra $\mathcal{C}$ is the subalgebra of endomorphisms of $H^{*}_{T}(\operatorname{X})$ generated by the endomorphisms ${\pi_{2}}_{*}(\alpha\cdot)\pi_{1}^{*}$ and ${\pi_{1}}_{*}(\alpha\cdot){\pi_{2}^{*}}$ for $\alpha\in H^{*}_{T}(\operatorname{X}_{(n,1,N-n-1)})$. In particular, $\mathcal{C}$ contains the following crucial elements,

[TABLE]

Definition 4.3**.**

Introduce the generating functions, for $0\leq n\leq N$,

[TABLE]

with the convention that for $n+1>N$ the corresponding operators are [math].

Definition 4.4**.**

The geometric Yang-Baxter algebras $\mathcal{YB}_{N}$ and $\mathcal{YB}^{\prime}_{N}$ are defined to be the subalgebras of $\mathcal{C}$ generated by the coefficients of $\mathcal{A}_{n}(x),\mathcal{B}_{n}(x),\mathcal{C}_{n}(x),\mathcal{D}_{n}(x)$, respectively of $\mathcal{A}^{\prime}_{n}(x),\mathcal{B}^{\prime}_{n}(x),\mathcal{C}^{\prime}_{n}(x),\mathcal{D}^{\prime}_{n}(x)$, where $0\leq n\leq N$.

Theorem 4.5**.**

(Geometric Yang-Baxter algebra) The algebras $\mathcal{YB}_{N}$ and $\mathcal{YB}^{\prime}_{N}$ are isomorphic to the Yang-Baxter algebras $\operatorname{YB}_{N}$ and $\operatorname{YB}_{N}^{\prime}$ respectively from Definition 2.2.

We will prove the theorem for the algebra $\mathcal{YB}_{N}$, the algebra $\mathcal{YB}^{\prime}_{N}$ can be treated similarly. We first check the last two relations listed in Proposition 2.1 for the geometric operators $\mathcal{A}(x),\mathcal{B}(x),\mathcal{C}(x),\mathcal{D}(x)$ and then identify them with the algebraic ones from Section 2. Finally we show hat this defines the operator $\mathcal{D}(x)$ uniquely, matching it with the algebraic operator $D(x)$ in Lemma 4.10. This will finish the proof at the end of the section.

4.3. Some relations

Checking the last two identities from (2.1) will be a standard fixed point calculation which goes back to the work of Atiyah and Bott [AB84, (3.8)]. In particular we need the Atiyah-Bott-Berline-Vergne integration formula. Let $X$ and $Y$ be compact non-singular algebraic $T$-varieties and $f:X\rightarrow Y$ be a $T$-equivariant morphism, then we have for all $\alpha\in H^{*}_{T}(X)$, (using notation from Definition 4.1 and the restriction $\operatorname{res}$ to fixed points,

[TABLE]

We apply this to the maps ${\pi_{1}}$ and ${\pi_{2}}$ from (4.6). The involved ratios of the Euler classes are

[TABLE]

The following explicitly computes the operators from Definition 4.3.

Proposition 4.6**.**

In the fixed point basis the following holds for $\operatorname{w}\in{\operatorname{X}}_{n}^{T}$, $z\in\overline{\operatorname{w}}$, and $z^{\prime}\in\operatorname{w}$.

1. 1.)

The operator $\mathcal{A}(x)$ is represented by a diagonal matrix with the eigenvalues

[TABLE] 2. 2.)

The $({\operatorname{w}}_{1}\cup z,\operatorname{w}_{1})$-matrix entry of $b_{n}$ is $\prod_{j\in{\operatorname{w}_{1}}}(t_{j}-t_{z})^{-1}$, the $({\operatorname{w}}_{1},{\operatorname{w}}_{1}\setminus z^{\prime})$-matrix entry of $c_{n}$ equals $\prod_{j\in\overline{{\operatorname{w}}_{1}}}(t_{z^{\prime}}-t_{j})^{-1}$. All other matrix entries are zero.

Proof.

By Lemma 4.2, the restriction of $c_{i}(\mathcal{T}_{n})$ (respectively $c_{i}(\mathcal{Q}_{n})$) to ${\operatorname{w}}$ is equal to the $i$th elementary symmetric function $\sigma_{i}$ in the variables $t_{j}$, $j\in{\operatorname{w}}_{1}$ (respectively $j\in\overline{\operatorname{w}_{1}}$). Thus (4.10) holds. If $\operatorname{w}_{1}$ denotes a fixed point in ${\operatorname{X}}_{n}^{T}$, then ${\pi_{1}}^{-1}(\operatorname{w}_{1})\cap\operatorname{X}_{(n,1,N-n-1)}=\{\operatorname{w}_{1}\cup z\mid z\in\overline{\operatorname{w}}_{1}\}$ and thus the asserted matrix coefficients ${\pi_{2}}_{*}{\pi_{1}^{*}}$ follow from (4.9), and similarly for ${\pi_{1}}_{*}{\pi_{2}^{*}}.$ ∎

Proposition 4.7**.**

The following identities hold

1. 1.)

$\mathcal{C}(x_{2})\mathcal{B}(x_{1})-\mathcal{C}(x_{1})\mathcal{B}(x_{2})=(x_{1}-x_{2})(\mathcal{A}(x_{1})\mathcal{D}(x_{2})-\mathcal{D}(x_{2})\mathcal{A}(x_{1})),$** 2. 2.)

$\mathcal{D}(x_{1})\mathcal{C}(x_{2})=\mathcal{D}(x_{2})\mathcal{C}(x_{1}),$** 3. 3.)

$\mathcal{C}(x_{1})\mathcal{D}(x_{2})-\mathcal{C}(x_{2})\mathcal{D}(x_{1})=(x_{1}-x_{2})\mathcal{D}(x_{2})\mathcal{C}(x_{1})$.

Proof.

We start with 1. The map $c_{n}b_{n}$ sends $\mathbbm{1}_{\operatorname{w}}$ to a linear combination of basis vectors, first via $b_{n}$ such vectors attached to fixed points of the form $\operatorname{w}_{1}\cup z$ for some $z\in\overline{\operatorname{w}_{1}}$ and then via $c_{n}$ to those of the form $(\operatorname{w}_{1}\cup z)\setminus z^{\prime}$, $z^{\prime}\in({\operatorname{w}_{1}}\cup z)$. In contrast, $b_{n}c_{n}$ first maps to those of the form $\operatorname{w}_{1}\setminus z^{\prime\prime}$ and then to $(\operatorname{w}_{1}\setminus z^{\prime\prime})\cup z^{\prime\prime\prime}$, where $z^{\prime\prime}\in\operatorname{w}_{1}$ and $z^{\prime\prime\prime}\in\overline{(\operatorname{w}_{1}\setminus z^{\prime\prime})}$. To compare, pick the same vector on each side. Assume first that $z=z^{\prime\prime\prime}\not=z^{\prime}=z^{\prime\prime}$. Then the denominator occurring in $c_{n}b_{n}$ is equal to ${\prod }_{j\in {\mathrm{w}}_{}}(t_j-t_z){\prod }_{j\in }$