# Modular Branching Rules for Projective Representations of Symmetric Groups and Lowering Operators for the Supergroup \(Q(n)\)

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*Alexander Kleshchev; Vladimir Shchigolev*

There are two approaches to projective representation theory of
symmetric and alternating groups, which are powerful enough to work for modular
representations. One is based on Sergeev duality, which connects projective
representation theory of the symmetric group and representation theory of the
algebraic supergroup \(Q(n)\) via appropriate Schur (super)algebras and
Schur functors. The second approach follows the work of Grojnowski for
classical affine and cyclotomic Hecke algebras and connects projective
representation theory of symmetric groups in characteristic \(p\) to the crystal
graph of the basic module of the twisted affine Kac-Moody algebra of type
\(A_{p-1}^{(2)}\).

The goal of this work is to connect the two approaches mentioned above and
to obtain new branching results for projective representations of symmetric
groups.

#### Table of Contents

# Table of Contents

## Modular Branching Rules for Projective Representations of Symmetric Groups and Lowering Operators for the Supergroup $Q(n)$

- Introduction vii8 free
- Set up vii8
- Projective representations and Sergeev algebra vii8
- Crystal graph approach ix10 free
- Schur functor approach xi12 free
- Modular branching rules xiii14 free
- Connecting the two approaches xiii14
- Some tensor products over 𝑄(𝑛) xiv15 free
- Strategy of the proof and organization of the paper xv16 free

- Chapter 1. Preliminaries 120 free
- Chapter 2. Lowering operators 1130
- Chapter 3. Some polynomials 2948
- Chapter 4. Raising coefficients 5170
- Chapter 5. Combinatorics of signature sequences 6180
- Chapter 6. Constructing 𝑈(𝑛-1)-primitive vectors 7594
- 6.1. Construction: case [∏_{𝑖<𝑘\𝑙𝑒𝑛}𝑟ᵦ(𝜆)_{𝑘}]=-^{𝑚} 7594
- 6.2. Construction: case [∏_{𝑖<𝑘<𝑛}𝑟ᵦ(𝜆)_{𝑘}]=-^{𝑚} and 𝜆ᵢ, 𝜆_{𝑛} are not both divisible by 𝑝 7897
- 6.3. Construction: case 𝜆ᵢ1\𝑝𝑚𝑜𝑑𝑝 and [∏_{𝑖<𝑘\𝑙𝑒𝑛}𝑟_{0}(𝜆)_{𝐤}]=+-^{𝐦} 81100
- 6.4. Extension: case 𝜆_{ℎ}0\𝑝𝑚𝑜𝑑𝑝, 𝜆ᵢ1\𝑝𝑚𝑜𝑑𝑝, [∏_{ℎ<𝑘\𝑙𝑒𝑖}𝑟_{0}(𝜆)_{𝐤}]=-^{𝐦} 85104
- 6.5. Extension: case 𝜆ᵢ̸0\𝑝𝑚𝑜𝑑𝑝, [∏_{ℎ<𝑘\𝑙𝑒𝑖}𝑟ᵦ(𝜆)_{𝑘}]=-^{𝑚}, and 𝜆_{ℎ}0\𝑝𝑚𝑜𝑑𝑝, 𝜆ᵢ1\𝑝𝑚𝑜𝑑𝑝 do not both hold 86105
- 6.6. Extension: case 𝜆_{ℎ}1\𝑝𝑚𝑜𝑑𝑝, 𝜆ᵢ0\𝑝𝑚𝑜𝑑𝑝, and [∏_{ℎ<𝑘\𝑙𝑒𝑖}𝑟_{0}(\𝐥𝐦)_{𝐤}]=+-^{𝐦} 89108

- Chapter 7. Main results on 𝑈(𝑛) 93112
- Chapter 8. Main results on projective representations of symmetric groups 113132
- Bibliography 121140