Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Pieri Rule at Infinity

Ivan Penkov, Pablo Zadunaisky

TL;DR

The Pieri rule at infinity studies the tensor product $oldsymbol{L}(oldsymbol{ heta}) ensor oldsymbol{F}$ for $ rak{gl}(oldsymbol{ ext{infty}})$, where $oldsymbol{L}(oldsymbol{ heta})$ is a simple integrable highest-weight module and $oldsymbol{F}$ is a simple weight-multiplicity-free module (including Fock and wedge/tensor types). The authors develop a linkage filtration governed by Pieri data and compatible orders, enabling a full description of simple constituents, a criterion for semisimplicity versus indecomposability, and a computation of socle and radical filtrations; they also prove rigidity under suitable finiteness hypotheses. The approach hinges on carefully chosen exhaustions of the index set by finite subalgebras and a detailed analysis of transition maps between finite-stage tensor products, generalizing classical Pieri behavior to the infinite setting. The results yield a robust framework for understanding indecomposable Pieri-type tensor categories in $ rak{gl}(oldsymbol{ extinfty})$, with implications for categorifications and O-type tensor categories in the infinite-dimension regime.

Abstract

We study the structure of tensor products of $\mathfrak{gl}(\infty) = \varinjlim \mathfrak{gl}(n)$-modules $\mathbf L(\mathbf λ) \otimes \mathbf F$ where $\mathbf L(\mathbf λ)$ is a simple integrable highest weight module and $\mathbf F$ is a simple integrable weight multiplicity-free module. Both $\mathbf L(\mathbf λ)$ and $\mathbf F$ are infinite dimensional, in particular $\mathbf F$ can be a Fock module. Similar tensor products of $\mathfrak{gl}(n)$-modules are semisimple and their simple constituents are described by the classical Pieri rule. We prove that a $\mathfrak{gl}(\infty)$-module $\mathbf M:= \mathbf L(\mathbf λ) \otimes \mathbf F$ is semisimple only in relatively trivial cases, and is indecomposable otherwise. Our main results are a description of the simple constituents of $\mathbf M$, and the construction of a linkage filtration on $\mathbf M$ that provides information on when two simple constituents of $\mathbf M$ are linked. Using the linkage filtration, we compute the socle and radical filtrations of $\mathbf M$, and determine when $\mathbf M$ is rigid.

The Pieri Rule at Infinity

TL;DR

The Pieri rule at infinity studies the tensor product for , where is a simple integrable highest-weight module and is a simple weight-multiplicity-free module (including Fock and wedge/tensor types). The authors develop a linkage filtration governed by Pieri data and compatible orders, enabling a full description of simple constituents, a criterion for semisimplicity versus indecomposability, and a computation of socle and radical filtrations; they also prove rigidity under suitable finiteness hypotheses. The approach hinges on carefully chosen exhaustions of the index set by finite subalgebras and a detailed analysis of transition maps between finite-stage tensor products, generalizing classical Pieri behavior to the infinite setting. The results yield a robust framework for understanding indecomposable Pieri-type tensor categories in , with implications for categorifications and O-type tensor categories in the infinite-dimension regime.

Abstract

We study the structure of tensor products of -modules where is a simple integrable highest weight module and is a simple integrable weight multiplicity-free module. Both and are infinite dimensional, in particular can be a Fock module. Similar tensor products of -modules are semisimple and their simple constituents are described by the classical Pieri rule. We prove that a -module is semisimple only in relatively trivial cases, and is indecomposable otherwise. Our main results are a description of the simple constituents of , and the construction of a linkage filtration on that provides information on when two simple constituents of are linked. Using the linkage filtration, we compute the socle and radical filtrations of , and determine when is rigid.
Paper Structure (38 sections, 32 theorems, 46 equations)

This paper contains 38 sections, 32 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Conventions on ordered sets
  4. The Lie algebra $\mathfrak{gl}(\infty)$
  5. Integrable highest weight $\mathfrak{gl}(\infty)$-modules
  6. Tensor modules
  7. Weight multiplicity-free $\mathfrak{gl}(\infty)$-modules
  8. Compatible orders
  9. Filtrations
  10. Main statements
  11. The sets $\mathbb J$ and $\mathbb J_{\infty}$
  12. Pieri sets and filtrations
  13. The statements
  14. Examples
  15. Tensor modules
  16. ...and 23 more sections

Key Result

Lemma 2.2

For $\boldsymbol{\lambda} \in \mathfrak{h}^*$ and $\mathbf F$ be as above, there exists a $(\boldsymbol{\lambda},\mathbf F)$-compatible total order $\preceq$. Furthermore, if $\mathbf L(\boldsymbol{\lambda})$ and $\mathbf F$ are both $\mathfrak{b}_{\preceq'}$-highest weight modules for some total or

Theorems & Definitions (76)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • ...and 66 more