scholarly journals TILING BRANCHING MULTIPLICITY SPACES WITH GL PATTERN BLOCKS

2013 ◽  
Vol 94 (3) ◽  
pp. 362-374 ◽  
Author(s):  
SANGJIB KIM
Keyword(s):  
Classical Groups ◽  
Orthogonal Groups ◽  
Branching Rules

AbstractWe study branching multiplicity spaces of complex classical groups in terms of ${\mathrm{GL} }_{2} $ representations. In particular, we show how combinatorics of ${\mathrm{GL} }_{2} $ representations are intertwined to make branching rules under the restriction of ${\mathrm{GL} }_{n} $ to ${\mathrm{GL} }_{n- 2} $. We also discuss analogous results for the symplectic and orthogonal groups.

scholarly journals A theorem on power series with applications to classical groups over finite fields

2001 ◽  
Vol 64 (1) ◽  
pp. 121-129
Author(s):  
Andrew J. Spencer
Keyword(s):  
Power Series ◽  
Finite Fields ◽  
Generating Functions ◽  
General Linear ◽  
Classical Groups ◽  
Orthogonal Groups

For some of the classical groups over finite fields it is possible to express the proportion of eigenvalue-free matrices in terms of generating functions. We prove a theorem on the monotonicity of the coefficients of powers of power series and apply this to the generating functions of the general linear, symplectic and orthogonal groups. This proves a conjecture on the monotonicity of the proportions of eigenvalue-free elements in these groups.

scholarly journals Factorial Characters and Tokuyama's Identity for Classical Groups

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Recurrence Relations ◽  
Function Theory ◽  
Lattice Paths ◽  
Schur Function ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Schubert Polynomial ◽  
International Audience ◽  
Symplectic Case ◽  
Polynomial Theory

International audience In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.

scholarly journals A reciprocal branching problem for automorphic representations and global Vogan packets

2020 ◽  
Vol 2020 (765) ◽  
pp. 249-277 ◽  
Author(s):  
Dihua Jiang ◽  
Baiying Liu ◽  
Bin Xu
Keyword(s):  
Irreducible Representation ◽  
Symmetry Breaking ◽  
Descent Method ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Automorphic Representations ◽  
Branching Rule ◽  
Branching Problem

AbstractLet G be a group and let H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π to H. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied to other classical groups as well.

scholarly journals ON INVOLUTIONS AND INDICATORS OF FINITE ORTHOGONAL GROUPS

2018 ◽  
Vol 105 (3) ◽  
pp. 380-416 ◽  
Author(s):  
GREGORY K. TAYLOR ◽  
C. RYAN VINROOT
Keyword(s):  
Asymptotic Behavior ◽  
Simple Group ◽  
Generating Functions ◽  
Finite Simple Group ◽  
Point Of View ◽  
Character Degree ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classical Groups ◽  
Orthogonal Groups

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

scholarly journals Pieri Rules for Classical Groups and Equinumeration between Generalized Oscillating Tableaux and Semistandard Tableaux

10.37236/6214 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Soichi Okada
Keyword(s):  
Classical Groups ◽  
Orthogonal Groups ◽  
Theoretic Proof ◽  
Pieri Rules

We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretic proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.

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