Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representation Theory of $0$-Schur Algebras and Related Categories

Woo-Seok Jung, Young-Tak Oh

Abstract

Jensen, Su, and Yang described the projective indecomposable modules of the $0$-Schur algebra $\mathbf{S}_0(n,r)$ using its geometric realization. In this paper, the simple modules of $\mathbf{S}_0(n,r)$ are identified by computing the tops of the projective indecomposable modules. Furthermore, functorial relations among the module categories $\mathbf{H}_r(0)$\textsf{-mod}, $\mathbf{S}_0(n,r)$\textsf{-mod}, and $U_0(\mathfrak{gl}_n)$\textsf{-mod} are examined, where $\mathbf{H}_r(0)$ denotes the $0$-Hecke algebra and $U_0(\mathfrak{gl}_n)$ denotes the degenerate quantum group.

Representation Theory of $0$-Schur Algebras and Related Categories

Abstract

Jensen, Su, and Yang described the projective indecomposable modules of the -Schur algebra using its geometric realization. In this paper, the simple modules of are identified by computing the tops of the projective indecomposable modules. Furthermore, functorial relations among the module categories \textsf{-mod}, \textsf{-mod}, and \textsf{-mod} are examined, where denotes the -Hecke algebra and denotes the degenerate quantum group.

Paper Structure

This paper contains 17 sections, 28 theorems, 212 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak and strong compositions
  4. The $0$-Schur algebra $S_0(n,r)$
  5. A $\mathbb Z$-basis and a presentation for $S_0(n,r)$
  6. Simple modules of the $0$-Schur algebra $\mathbf{S}_0(n, r)$
  7. Projective indecomposable $\mathbf{S}_0(n, r)$-modules
  8. Construction of simple $\mathbf{S}_0(n, r)$-modules
  9. Functorial Relations Among Module Categories
  10. The $0$-Hecke algebra $\mathbf{H}_r(0)$ and its representation theory
  11. The degenerate quantum group $U_0(\mathfrak{gl}_n)$ and its polynomial modules
  12. Functorial relations among $U_0(\mathfrak{gl}_n)$-mod, $\mathbf{S}_0(n,r)$-mod, and $\mathbf{H}_r(0)$-mod
  13. Projectivity of $G_{n,r}(R_{\alpha})$
  14. Images of simple and projective indecomposable $\mathbf{H}_r(0)$-modules under the functor $G_{n,r}$
  15. Proof of \ref{['lem: phi']}
  16. ...and 2 more sections

Key Result

Lemma 2.1

(JS15) Let $A \in M_n(r)$ with $\mathsf{row}(A)=(\lambda_1, \lambda_2,\ldots, \lambda_n)$.

Figures (2)

  • Figure 3.1: The $S_0(3,3)_{\mathbb{C}}$-action on $P_{(2,1,0)}$
  • Figure 3.2: The $S_0(3,3)_{\mathbb{C}}$-action on $S_{(2,1,0)}$

Theorems & Definitions (56)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Example 3.3
  • Definition 3.4
  • Example 3.5
  • Proposition 3.6
  • proof
  • Example 3.7
  • ...and 46 more