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Nil graded algebras associated to triangular matrices and their applications to Soergel Calculus

Diego Lobos

Abstract

We introduce and study a category of algebras strongly connected with the structure of the Gelfand-Tsetlin subalgebras of the endomorphism algebras of Bott-Samelson bimodules. We develop a series of techniques that allow us to obtain optimal presentations for the many Gelfand-Tsetlin subalgebras appearing in the context of the Diagrammatic Soergel Category.

Nil graded algebras associated to triangular matrices and their applications to Soergel Calculus

Abstract

We introduce and study a category of algebras strongly connected with the structure of the Gelfand-Tsetlin subalgebras of the endomorphism algebras of Bott-Samelson bimodules. We develop a series of techniques that allow us to obtain optimal presentations for the many Gelfand-Tsetlin subalgebras appearing in the context of the Diagrammatic Soergel Category.
Paper Structure (14 sections, 25 theorems, 107 equations, 1 figure)

This paper contains 14 sections, 25 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Notations and terminology
  4. Acknowledgment
  5. Nil graded algebras associated to triangular matrices
  6. Nil graded algebras weakly associated to triangular matrices
  7. Nil graded algebras strongly associated to triangular matrices
  8. The category $\boldsymbol{\mathcal{NT}}.$
  9. The $\nabla-$products in $\boldsymbol{\mathcal{NT}}$
  10. Applications to Soergel Calculus.
  11. The Diagrammatic Soergel category
  12. Nil graded algebras associated to free expressions
  13. Gelfand-Tsetlin subalgebras
  14. Explicit calculus in type $\widetilde{A}$

Key Result

Lemma 2.3

Let $\mathcal{A}$ an algebra weakly associated to a strictly lower triangular matrix $T=[t_{ij}]_{n\times n},$ relative to a set of generators $\mathcal{X}=\{X_1,\dots,X_n\}.$ Then the vector space structure of $\mathcal{A},$ is generated by the set: (note that $1=X_1^{0}X_2^{0}\cdots X_{n}^{0}\in W_{0}$).

Figures (1)

  • Figure 1: A Soergel graph

Theorems & Definitions (68)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 58 more