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

Intro -- Contents -- Abstract -- Introduction -- Set up -- Projective representations and Sergeev algebra -- Crystal graph approach -- Schur functor approach -- Modular branching rules -- Connecting the two approaches -- Some tensor products over ( ) -- Strategy of the proof and organization of the paper -- Chapter 1. Preliminaries -- 1.1. General Notation -- 1.2. The supergroup ( ) and its hyperalgebra -- 1.3. Highest weight theory -- Chapter 2. Lowering operators -- 2.1. Definitions -- 2.2. Properties of ^{ }_{ , }({ }) and ^{ }_{ , }({ }) -- 2.3. Supercommutator [ _{ }^{ }, _{ , }^{ }( )] -- 2.4. Supercommutator [ ⱼ^{ }, _{ , }^{ }( )] -- 2.5. More on _{ }^{ } _{ , }^{ }( ) -- 2.6. Some coefficients -- Chapter 3. Some polynomials -- 3.1. Operators ^{ }_{ , } -- 3.2. Polynomials _{ , }^{ , }( ) -- 3.3. Polynomials ⁽¹⁾_{ , }( ) -- 3.4. Polynomials ⁽²⁾_{ , , , }( ) -- Chapter 4. Raising coefficients -- 4.1. Inductive formulas -- 4.2. The case of signed sets with only even elements -- 4.3. The case of signed sets with one odd element -- Chapter 5. Combinatorics of signature sequences -- 5.1. Marked signature sequences -- 5.2. Normal and good indices -- 5.3. Tensor conormal and tensor cogood indices -- 5.4. Removable and addable nodes for dominant -strict weights -- Chapter 6. Constructing ( -1)-primitive vectors -- 6.1. Construction: case [∏_{ &lt -- \ } ᵦ( )_{ }]=-^{ } -- 6.2. Construction: case [∏_{ &lt -- &lt -- } ᵦ( )_{ }]=-^{ } and ᵢ, _{ } are not both divisible by -- 6.3. Construction: case ᵢ1\ and [∏_{ &lt -- \ } _{0}( )_{ }]=+-^{ } -- 6.4. Extension: case _{ℎ}0\ , ᵢ1\ , [∏_{ℎ&lt -- \ } _{0}( )_{ }]=-^{ } -- 6.5. Extension: case ᵢ̸0\ , [∏_{ℎ&lt.

\ } ᵦ( )_{ }]=-^{ }, and _{ℎ}0\ , ᵢ1\ do not both hold -- 6.6. Extension: case _{ℎ}1\ , ᵢ0\ , and [∏_{ℎ&lt -- \ } _{0}(\ )_{ }]=+-^{ } -- Chapter 7. Main results on ( ) -- 7.1. Normal indices and primitive vectors -- 7.2. Criterion for existence of nonzero ( -1)-primitive vectors -- 7.3. The socle of the first level -- 7.4. Complement pairs -- 7.5. Primitive vectors in ( )⊗ * -- 7.6. Primitive vectors in ( )⊗ -- Chapter 8. Main results on projective representations of symmetric groups -- 8.1. Representations of Sergeev superalgebras -- 8.2. Proof of Theorem A -- 8.3. Proof of Theorem B -- 8.4. Projective representations of symmetric groups -- Bibliography.

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