Schubert Derivations on the Infinite Wedge Power

TL;DR

This paper extends Schubert derivations to the infinite exterior power of a free abelian group, linking them to vertex operators and the bosonic vertex representation of $gl_00$, revealing new algebraic structures in infinite-dimensional settings.

Contribution

It introduces a novel extension of Schubert derivations to infinite wedge spaces and connects classical vertex operators with the cohomology of Grassmannians within this framework.

Findings

Derived vertex operators from the integration by parts formula.

Connected Schubert calculus with the bosonic vertex representation.

Interpreted DJKM result as a limit case in infinite-dimensional cohomology.

Abstract

The {\em Schubert derivation} is a distinguished Hasse-Schmidt derivation on the exterior algebra of a free abelian group, encoding the formalism of Schubert calculus for all Grassmannians at once. The purpose of this paper is to extend the Schubert derivation to the infinite exterior power of a free ${\mathbb Z}$-module of infinite rank (fermionic Fock space). Classical vertex operators naturally arise from the {\em integration by parts formula}, that also recovers the generating function occurring in the {\em bosonic vertex representation} of the Lie algebra $gl_\infty({\mathbb Z})$, due to Date, Jimbo, Kashiwara and Miwa (DJKM). In the present framework, the DJKM result will be interpreted as a limit case of the following general observation: the singular cohomology of the complex Grassmannian $G(r,n)$ is an irreducible representation of the Lie algebra of $n\times n$ square matrices.}

Equations

$$g{l}_n(\mathbb{Z}):=\underset{0\le i,j<n}{⨁}\mathbb{Z}\cdot {\mathcal{B}}_{ij}\subseteq {\mathrm{End}}_{\mathbb{Z}}(M_n),$$

$$\{\begin{array}{ccccc}\delta (A)\mathbf{u}& :=& A\mathbf{u},& & \forall \mathbf{u}\in M_n\\ & & & & \\ \delta (A)(\mathbf{v}\wedge \mathbf{w})& =& \delta (A)\mathbf{v}\wedge \mathbf{w}+\mathbf{v}\wedge \delta (A)\mathbf{w},& & \forall \mathbf{v},\mathbf{w}\in \bigwedge M_n.\end{array}$$

Full text

*To Israel Vainsencher on the occasion of his seventieth birthday          *

1. Introduction

Introduction

0.1

The Goal. Let $r,n\in\mathbb{N}\cup\{\infty\}$ such that $r\leq n$. The main characters of this paper are i) the exterior algebra $\bigwedge M_{n}$ of a free abelian group $M_{n}:=\bigoplus_{0\leq j<n}\mathbb{Z}b_{j}$ and ii) the cohomology ring $B_{r,n}$ of the Grassmann variety $G(r,n)$. By the latter we mean the following. If $0\leq r\leq n<\infty$, $B_{r,n}$ stands for the usual singular cohomology ring $H^{*}(G(r,n),\mathbb{Z})$ of the Grasmannian variety parametrizing $r$-dimensional subspaces of the complex $n$-dimensional vector space. If $n=\infty$, the ring $B_{r}:=B_{r,\infty}$ will denote the cohomology of the ind-variety $G(r,\infty)$ (see e.g. [bottu, p. 302] or [DimiPenkov, IgnaPenkov]), which is a polynomial ring $\mathbb{Z}[e_{1},\ldots,e_{r}]$ in $r$-indeterminates. If $r=n=\infty$, instead $Gr(\infty):=G(\infty,\infty)$ is the ind-Grassmannian constructed e.g. in [IgnaPenkov, Section 3.3] or the Sato’s Universal Grassmann Manifold (UGM), as in e.g. [satoUGM]. In this case $B:=B_{\infty,\infty}$ is the $\mathbb{Z}$-polynomial ring in infinitely many indeterminates. Let ${\mathcal{B}}_{ij}\in\mathrm{End}_{\mathbb{Z}}(M_{n})$ such that ${\mathcal{B}}_{ij}(b_{k})=b_{i}\delta_{jk}$, and let

[TABLE]

which is a Lie algebra with respect to the usual commutator. Clearly ${gl_{n}(\mathbb{Z})}=\mathrm{End}_{\mathbb{Z}}(M_{n})$ if $n<\infty$. This paper is inspired by the following simple observation, for which we have not been able to find an explicit reference in the literature:

The ring $B_{r,n}$ is a module over the Lie algebra ${gl_{n}(\mathbb{Z})}$.

If $r=1$ the claim is obvious, because if $e$ denotes the hyperplane class of $\mathbb{P}^{n-1}$, then $B_{1,n}=\mathbb{Z}[e]/(e^{n})$ is a free abelian group of rank $n$ and, therefore, the standard representation of its Lie algebra of endomorphisms. The general case, for $r<\infty$ and arbitrary $n\geq r$, follows from noticing that all $A\in{gl_{n}(\mathbb{Z})}$ induce an even derivation $\delta(A)$ on ${\bigwedge M_{n}}$:

[TABLE]

Since $\delta([A,B])=[\delta(A),\delta(B)]$, the map $A\mapsto\delta(A)_{|\bigwedge^{r}M}$ makes $\bigwedge^{r}M_{n}$ into an (irreducible) representation of ${gl_{n}(\mathbb{Z})}$. Then $B_{r,n}$ gets equipped with a ${gl_{n}(\mathbb{Z})}$-module structure as well, due to the $\mathbb{Z}$-module isomorphism $B_{r,n}\rightarrow\bigwedge^{r}M_{n}$. Recall that the latter is the composition of the Poincaré isomorphism, mapping $B_{r,n}$ onto its singular homology $H_{*}(G(r,n),\mathbb{Z})$, with the natural isomorphism $H_{*}(G(r,n),\mathbb{Z})\rightarrow\bigwedge^{r}H_{*}(\mathbb{P}^{n-1},\mathbb{Z})\cong\bigwedge^{r}M_{n}$, as in [SCHSD], or [GatSal4, diagramme (5.27)].

For $r=\infty$, the fact that $B$ is a ${gl_{\infty}(\mathbb{Z})}$-module is well known, and is due to the isomorphism of $B$ with each degree of the fermionic Fock space which, roughly speaking, plays the role of an infinite exterior power. Its structure has been explicitly described by Date, Jimbo, Kashiwara and Miwa (DJKM) in [DJKM01], see also [jimbomiwa, Formula (1.17)] and [KacRaRoz, p. 53], by computing the shape of a generating function ${\mathcal{B}}(z,w)$ encoding the multiplication of any polynomial by elementary matrices ${\mathcal{B}}_{ij}$ of infinite sizes.

In our contribution [GatSal6] we determine the shape of the same generating function in the case $r<\infty$, by using the formalism of Schubert derivations in the sense of [GatSal4]. The formula we obtain is new (as far as we know), and has a classical flavor (occurring as a $2$-parameter deformation of the Schur determinant occurring in Giambelli’s formula). We then felt the need to show that our methods also work in the known case $r=\infty$. The output is the present paper, in which we offer an alternative deduction of the DJKM bosonic vertex representation of ${gl_{\infty}(\mathbb{Z})}$, based on the extension of the Schubert derivations to an infinite wedge power. Our method to compute the ${gl_{n}(\mathbb{Z})}$-structure of $B_{r,n}$ then works uniformly for all pairs $r\leq n$ ranging over $\mathbb{N}\cup\{\infty\}$.