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Gelfand-Tsetlin modules: canonicity and calculations

Turner Silverthorne, Ben Webster

Abstract

In this paper, we give a more down-to-earth introduction to the connection between Gelfand-Tsetlin modules over $\mathfrak{gl}_n$ and diagrammatic KLRW algebras, and develop some of its consequences. In addition to a new proof of this description of the category Gelfand-Tsetlin modules appearing in earlier work, we show three new results of independent interest: (1) we show that every simple Gelfand-Tsetlin module is a canonical module in the sense of Early, Mazorchuk and Vishnyakova, and characterize when two maximal ideals have isomorphic canonical modules, (2) we show that the dimensions of Gelfand-Tsetlin weight spaces in simple modules can be computed using an appropriate modification of Leclerc's algorithm for computing dual canonical bases, and (3) we construct a basis of the Verma modules of $\mathfrak{sl}_n$ which consists of generalized eigenvectors for the Gelfand-Tsetlin subalgebra. Furthermore, we present computations of multiplicities and Gelfand-Kirillov dimensions for all integral Gelfand-Tsetlin modules in ranks 3 and 4; unfortunately, for ranks $>4$, our computers are not adequate to perform these computations.

Gelfand-Tsetlin modules: canonicity and calculations

Abstract

In this paper, we give a more down-to-earth introduction to the connection between Gelfand-Tsetlin modules over and diagrammatic KLRW algebras, and develop some of its consequences. In addition to a new proof of this description of the category Gelfand-Tsetlin modules appearing in earlier work, we show three new results of independent interest: (1) we show that every simple Gelfand-Tsetlin module is a canonical module in the sense of Early, Mazorchuk and Vishnyakova, and characterize when two maximal ideals have isomorphic canonical modules, (2) we show that the dimensions of Gelfand-Tsetlin weight spaces in simple modules can be computed using an appropriate modification of Leclerc's algorithm for computing dual canonical bases, and (3) we construct a basis of the Verma modules of which consists of generalized eigenvectors for the Gelfand-Tsetlin subalgebra. Furthermore, we present computations of multiplicities and Gelfand-Kirillov dimensions for all integral Gelfand-Tsetlin modules in ranks 3 and 4; unfortunately, for ranks , our computers are not adequate to perform these computations.

Paper Structure

This paper contains 25 sections, 37 theorems, 79 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Gelfand-Tsetlin modules
  4. OGZ algebras
  5. Completion and Gelfand-Tsetlin modules
  6. KLRW algebras
  7. Polynomial representation
  8. Induction
  9. Thick calculus
  10. Standard modules
  11. Computation of characters of simple modules
  12. Canonical modules
  13. Gelfand-Tsetlin modules
  14. Equivalence to KLR
  15. The non-integral case
  16. ...and 10 more sections

Key Result

Theorem A

Figures (8)

  • Figure 1: Basis vectors for simple $\tilde{\mathbb{T}}$-modules, part I
  • Figure 2: Basis vectors for simple $\tilde{\mathbb{T}}$-modules, part II
  • Figure 3: The table of dimensions of GT weight spaces in the integral regular case for $\mathfrak{gl}_3$
  • Figure 4: The characters of GT modules when $\iota_1=1,\iota_3=2$ (case (C5) in the notation of futornyGelfandTsetlinModules2018)
  • Figure 5: Similarly, here are the tables for (1) $\iota_1=\iota_3=1$ (C4) (2) $\iota_1=0,\iota_3=2$ (C6), and (3) $\iota_1=0, \iota_3=3$ (C9), (4) $\iota_1=1,\iota_3=0$ (C2), (5) $\iota_1=\iota_3=0$ (C1); the case $\iota_1=0,\iota_3=1$ (C3) is identical to (4).
  • ...and 3 more figures

Theorems & Definitions (72)

  • Theorem A
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 62 more