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Simple character formulas for finite $W$-superalgebras of type $A$

Shun-Jen Cheng, Weiqiang Wang

Abstract

We establish a canonical basis character formula for the irreducible modules in arbitrary parabolic BGG-type categories, including the category of finite-dimensional modules, for finite $W$-superalgebras of type $A$. These categories categorify the tensor product modules of irreducible polynomial representations and their duals over a quantum group of type $A$. Moreover, the standard modules and irreducible modules in these categories categorify the standard basis and Lusztig's dual canonical basis in the tensor product modules. Our formula provides a uniform generalization of several character formulas in BGG categories for Lie superalgebras and for $W$-algebras of type $A$.

Simple character formulas for finite $W$-superalgebras of type $A$

Abstract

We establish a canonical basis character formula for the irreducible modules in arbitrary parabolic BGG-type categories, including the category of finite-dimensional modules, for finite -superalgebras of type . These categories categorify the tensor product modules of irreducible polynomial representations and their duals over a quantum group of type . Moreover, the standard modules and irreducible modules in these categories categorify the standard basis and Lusztig's dual canonical basis in the tensor product modules. Our formula provides a uniform generalization of several character formulas in BGG categories for Lie superalgebras and for -algebras of type .
Paper Structure (24 sections, 15 theorems, 97 equations)

This paper contains 24 sections, 15 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. The goal
  3. Dual canonical and standard bases
  4. Simple character formulas
  5. Relations to other works
  6. Organization
  7. Dual canonical and standard bases on tensor product modules
  8. Tableaux
  9. Bruhat orderings
  10. Tensor $\mathbf{U}$-modules $\mathbb{T}^{\mathbf{s}}$
  11. $q$-symmetric tensors
  12. $q$-exterior tensors
  13. The $\mathbf{U}$-modules $P^{\boldsymbol{\epsilon\lambda}} (\mathbb V)$
  14. Partial ordering on $\text{Std}(\boldsymbol{\epsilon\lambda})$
  15. Dual canonical basis for $\widehat{P}^{\boldsymbol{\epsilon\lambda}} (\mathbb V)$
  16. ...and 9 more sections

Key Result

Theorem A

Theorems & Definitions (31)

  • Theorem A: Theorem \ref{['thm:DCB:Slambda']}, Theorem \ref{['thm:DCB_P']}, Proposition \ref{['prop:sameDCB']}
  • Theorem B: Theorem \ref{['thm:char']}
  • Theorem C: Theorem \ref{['thm:charStandard']}
  • Theorem D: Theorem \ref{['thm:fd_Char']}
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Example 2.5
  • ...and 21 more