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On dimensions of RoCK blocks of cyclotomic quiver Hecke superalgebras

Alexander Kleshchev

Abstract

We explicitly compute the dimensions of certain idempotent truncations of RoCK blocks of cyclotomic quiver Hecke superalgebras. Equivalently, this amounts to a computation of the value of the Shapovalov form on certain explicit vectors in the basic representations of twisted affine Kac-Moody Lie algebras of type $A$.

On dimensions of RoCK blocks of cyclotomic quiver Hecke superalgebras

Abstract

We explicitly compute the dimensions of certain idempotent truncations of RoCK blocks of cyclotomic quiver Hecke superalgebras. Equivalently, this amounts to a computation of the value of the Shapovalov form on certain explicit vectors in the basic representations of twisted affine Kac-Moody Lie algebras of type .

Paper Structure

This paper contains 16 sections, 18 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Shapovalov forms
  3. Lie theoretic notation
  4. Quantized enveloping algebra
  5. Putting $q$ to $1$
  6. Combinatorics
  7. Partitions, multipartitions, tableaux
  8. Addable and removable nodes
  9. Symmetric functions
  10. Another description of $\Pi_{\mu,\text{\boldmath$j$}}$
  11. Computing the inner product $(\Pi_{\mu,\text{\boldmath$j$}},\Pi_{\omega_d})_{\mathsf{Sym}}$
  12. Fock space
  13. Fock spaces ${\mathscr F}_q$ and ${\mathscr F}$
  14. The elements $\chi_\lambda$
  15. Shapovalov form for RoCK weights
  16. ...and 1 more sections

Key Result

Lemma 2.2

Let $\Lambda\in P_+$ and $w\in W$ with a reduced decomposition $w=r_{i_t}\cdots r_{i_1}$. Then:

Theorems & Definitions (36)

  • Lemma 2.2
  • proof
  • Lemma 2.5
  • Lemma 3.7
  • Example 3.8
  • Example 3.19
  • Example 3.28
  • Lemma 3.30
  • proof
  • Corollary 3.32
  • ...and 26 more