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Simple subquotients of relation modules

Gustavo Costa, Lucas Queiroz Pinto, Luis Enrique Ramirez

TL;DR

The paper addresses the problem of explicitly describing simple subquotients of relation Gelfand–Tsetlin modules for $\mathfrak{gl}(n)$. It develops a graph-based framework using relation graphs $G$ and tableau realizations to construct submodules and their irreducible quotients, with key tools including $G$-realizations, the sets $\Omega^{+}(T(Q))$, maximal $G$-chains, and a partial order on tableaux. The main contributions are explicit bases for the submodule generated by a tableau and for the corresponding simple subquotient, namely $\mathcal{N}_G(T(R))$ and $\mathcal{I}_G(T(R))$, and the quotient construction $M(T(R))$, together with a generalization of prior work (notably $FGR15$) to non-generic settings. The results unify and extend known GT-module constructions (finite-dimensional, generic, cuspidal) and provide a systematic method to classify simple constituents via combinatorial data encoded in $E^+_{G(R)}$. This advances understanding of GT representations by delivering explicit, computable bases for simple subquotients and linking combinatorial graphs to module structure.

Abstract

In this paper we provide an explicit tableaux realization for all simple subquotients of a relation Gelfand-Tsetlin $\mathfrak{gl}(n)$-module.

Simple subquotients of relation modules

TL;DR

The paper addresses the problem of explicitly describing simple subquotients of relation Gelfand–Tsetlin modules for . It develops a graph-based framework using relation graphs and tableau realizations to construct submodules and their irreducible quotients, with key tools including -realizations, the sets , maximal -chains, and a partial order on tableaux. The main contributions are explicit bases for the submodule generated by a tableau and for the corresponding simple subquotient, namely and , and the quotient construction , together with a generalization of prior work (notably ) to non-generic settings. The results unify and extend known GT-module constructions (finite-dimensional, generic, cuspidal) and provide a systematic method to classify simple constituents via combinatorial data encoded in . This advances understanding of GT representations by delivering explicit, computable bases for simple subquotients and linking combinatorial graphs to module structure.

Abstract

In this paper we provide an explicit tableaux realization for all simple subquotients of a relation Gelfand-Tsetlin -module.

Paper Structure

This paper contains 7 sections, 10 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gelfand-Tsetlin modules
  3. Relation Gelfand-Tsetlin modules
  4. Relation graphs
  5. Useful definitions
  6. Simple subquotients
  7. Examples

Key Result

Theorem 2.3

Let $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ be an integral dominant $\mathfrak{gl}(n)$-weight (i.e., $\lambda_i-\lambda_{i+1}\in\mathbb{Z}_{\ge 0}$, for all $1\leq i\leq n-1$). The simple finite-dimensional module $L(\lambda)$ has a realization of tableaux, where the vector space consists of all where $T(L\pm\delta^{ki})$ is the tableau obtained from $T(L)$ by adding $\pm 1$ to the $(k, i)$'s

Figures (1)

  • Figure 1: Configuration of the set of vertices $\mathfrak{V}$ for $n=4$

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Gelfand-Tsetlin-1950
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Theorem 3.4
  • Definition 3.5
  • Example 3.6
  • Remark 3.7
  • ...and 25 more