# modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup

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## Modular Branching Rules For Projective Representations Of Symmetric Groups And Lowering Operators For The Supergroup Q N

**Author :**Aleksandr Sergeevich Kleshchëv

**ISBN :**9780821874318

**Genre :**Mathematics

**File Size :**57. 11 MB

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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.

## The Poset Of K Shapes And Branching Rules For K Schur Functions

**Author :**Thomas Lam

**ISBN :**9780821872949

**Genre :**Mathematics

**File Size :**89. 27 MB

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The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.

## Isolated Involutions In Finite Groups

**Author :**Rebecca Waldecker

**ISBN :**9780821888032

**Genre :**Mathematics

**File Size :**26. 41 MB

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This text provides a new proof of Glauberman's Z*-Theorem under the additional hypothesis that the simple groups involved in the centraliser of an isolated involution are known simple groups.

## Torsors Reductive Group Schemes And Extended Affine Lie Algebras

**Author :**Philippe Gille

**ISBN :**9780821887745

**Genre :**Mathematics

**File Size :**86. 91 MB

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The authors give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that the authors take draws heavily from the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows the authors to find a bridge between multiloop algebras and the work of F. Bruhat and J. Tits on reductive groups over complete local fields.

## Pseudo Differential Operators With Discontinuous Symbols Widom S Conjecture

**Author :**Aleksandr Vladimirovich Sobolev

**ISBN :**9780821884874

**Genre :**Mathematics

**File Size :**71. 95 MB

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Relying on the known two-term quasiclassical asymptotic formula for the trace of the function $f(A)$ of a Wiener-Hopf type operator $A$ in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalization of that formula for a pseudo-differential operator $A$ with a symbol $a(\mathbf{x}, \boldsymbol{\xi})$ having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper provides a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.

## Non Cooperative Equilibria Of Fermi Systems With Long Range Interactions

**Author :**Jean-Bernard Bru

**ISBN :**9780821889763

**Genre :**Mathematics

**File Size :**71. 84 MB

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The authors define a Banach space $\mathcal{M}_{1}$ of models for fermions or quantum spins in the lattice with long range interactions and make explicit the structure of (generalized) equilibrium states for any $\mathfrak{m}\in \mathcal{M}_{1}$. In particular, the authors give a first answer to an old open problem in mathematical physics--first addressed by Ginibre in 1968 within a different context--about the validity of the so-called Bogoliubov approximation on the level of states. Depending on the model $\mathfrak{m}\in \mathcal{M}_{1}$, the authors' method provides a systematic way to study all its correlation functions at equilibrium and can thus be used to analyze the physics of long range interactions. Furthermore, the authors show that the thermodynamics of long range models $\mathfrak{m}\in \mathcal{M}_{1}$ is governed by the non-cooperative equilibria of a zero-sum game, called here thermodynamic game.

## On Some Aspects Of Oscillation Theory And Geometry

**Author :**Bruno Bianchini

**ISBN :**9780821887998

**Genre :**Mathematics

**File Size :**72. 1 MB

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The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation the authors prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep their investigation basically self-contained, the authors also collect some, more or less known, material which often appears in the literature in various forms and for which they give, in some instances, new proofs according to their specific point of view.