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Branching algebras for the general linear Lie superalgebra

Soo Teck Lee, Ruibin Zhang

Abstract

We develop an algebraic approach to the branching of representations of the general linear Lie superalgebra $\mathfrak{gl}_{p|q}({\mathbb C})$, by constructing certain super commutative algebras whose structure encodes the branching rules. Using this approach, we derive the branching rules for restricting any irreducible polynomial representation $V$ of $\mathfrak{gl}_{p|q}({\mathbb C})$ to a regular subalgebra isomorphic to $\mathfrak{gl}_{r|s}({\mathbb C})\oplus \mathfrak{gl}_{r'|s'}({\mathbb C})$, $\mathfrak{gl}_{r|s}({\mathbb C})\oplus\mathfrak{gl}_1({\mathbb C})^{r'+s'}$ or $\mathfrak{gl}_{r|s}({\mathbb C})$, with $r+r'=p$ and $s+s'=q$. In the case of $\mathfrak{gl}_{r|s}({\mathbb C})\oplus\mathfrak{gl}_1({\mathbb C})^{r'+s'}$ with $s=0$ or $s=1$ but general $r$, we also construct a basis for the space of $\mathfrak{gl}_{r|s}({\mathbb C})$ highest weight vectors in $V$; when $r=s=0$, the branching rule leads to explicit expressions for the weight multiplicities of $V$ in terms of Kostka numbers.

Branching algebras for the general linear Lie superalgebra

Abstract

We develop an algebraic approach to the branching of representations of the general linear Lie superalgebra , by constructing certain super commutative algebras whose structure encodes the branching rules. Using this approach, we derive the branching rules for restricting any irreducible polynomial representation of to a regular subalgebra isomorphic to , or , with and . In the case of with or but general , we also construct a basis for the space of highest weight vectors in ; when , the branching rule leads to explicit expressions for the weight multiplicities of in terms of Kostka numbers.
Paper Structure (21 sections, 9 theorems, 120 equations)

This paper contains 21 sections, 9 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $({\mathfrak{gl}}_n, {\mathfrak{gl}}_{p|q})$ Howe duality
  4. Linear algebra of ${\mathbb Z}_2$-graded vector spaces
  5. The general linear Lie superalgebra ${\mathfrak{gl}}_{p|q}$
  6. Realization of irreducible polynomial representations of ${\mathfrak{gl}}_{p|q}$
  7. Iterated Pieri rules for ${\mathfrak{gl}}_n$
  8. Branching algebra for $({\mathfrak{gl}}_{p|q},{\mathfrak{gl}}_{r|s}\oplus \mathfrak{h}_{r'}\oplus\mathfrak{h}_{s'})$
  9. Branching problems
  10. The branching algebra $\mathcal{Q}(\mathfrak{g},\mathfrak{m})$
  11. Branching rules for $(\mathfrak{g},\mathfrak{m})$ and $(\mathfrak{g},\mathfrak{g}')$
  12. Weight multiplicities of $L_{p|q}^F$
  13. Branching algebra for $({\mathfrak{gl}}_{p|q},{\mathfrak{gl}}_{r|s}\oplus{\mathfrak{gl}}_{p-r|q-s})$
  14. Branching problem
  15. The branching algebra $\mathcal{Q}(\mathfrak{g},\mathfrak{q})$
  16. ...and 6 more sections

Key Result

Theorem 2.9

As a ${\mathfrak{gl}}_n\oplus\mathfrak{gl}_{p|q}$-module, $\mathcal{R}$ has the following multiplicity free decomposition into irreducible represenations where $\Lambda^{+}_{n,p|q}=\Lambda^{+}_n\cap \Lambda^{+}_{p|q},$ and $\rho^F_n$ is the irreducible ${\mathfrak{gl}}_n$ representation with highest weight determined by $F$ in the usual way (see Remark rmk:hw below).

Theorems & Definitions (32)

  • Remark 2.5
  • Remark 2.8
  • Theorem 2.9: The $({\mathfrak{gl}}_n, {\mathfrak{gl}}_{p|q})$-duality
  • Remark 2.11
  • Proposition 2.17: Iterated Pieri rules
  • proof
  • Proposition 3.9
  • proof
  • Remark 3.17
  • Proposition 3.18
  • ...and 22 more