scholarly journals Schur $q$-functions and Spin Characters of Symmetric Groups I

10.37236/1278 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Alun Morris ◽  
A. A. Abdel-Aziz
Keyword(s):  
Symmetric Groups ◽  
Schur Functions ◽  
Irreducible Characters ◽  
Spin Characters

Inspired by the early work of D.E.Littlewood and A.R.Richardson, Schur functions have been used to give useful combinatorial formulae for determining explicit values for the irreducible characters of the symmetric groups.In this,the first of two papers,we consider how Schur Q-functions can be used to obtain combinatorial formulae for the irreducible spin (projective) characters of symmetric groups.

Orthogonal $*$-basis of relative symmetry classes of polynomials

2015 ◽  
Vol 30 ◽  
Author(s):  
Kijti Rodtes
Keyword(s):  
Finite Groups ◽  
Symmetric Groups ◽  
Orthogonal Basis ◽  
Irreducible Characters ◽  
Symmetry Classes

In this note, the existence of orthogonal ∗-basis of the symmetry classes of polynomials is discussed. Analogously to the orthogonal ∗-basis of symmetry classes of tensor, some criteria for the existence of the basis for finite groups are provided. A condition for the existence of such basis of symmetry classes of polynomials associated to symmetric groups and some irreducible characters is also investigated.

scholarly journals Spin characters of generalized symmetric groups

2013 ◽  
Vol 173 (4) ◽  
pp. 495-518 ◽  
Author(s):  
Xiaoli Hu ◽  
Naihuan Jing
Keyword(s):  
Symmetric Groups ◽  
Spin Characters

scholarly journals Asymptotics of Young Diagrams and Hook Numbers

10.37236/1307 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Amitai Regev ◽  
Anatoly Vershik
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Methods ◽  
Symmetric Groups ◽  
Schur Functions ◽  
Young Diagrams ◽  
Arithmetic Properties ◽  
Hook Formula ◽  
Hook Numbers

Asymptotic calculations are applied to study the degrees of certain sequences of characters of symmetric groups. Starting with a given partition $\mu$, we deduce several skew diagrams which are related to $\mu$. To each such skew diagram there corresponds the product of its hook numbers. By asymptotic methods we obtain some unexpected arithmetic properties between these products. The authors do not know "finite", nonasymptotic proofs of these results. The problem appeared in the study of the hook formula for various kinds of Young diagrams. The proofs are based on properties of shifted Schur functions, due to Okounkov and Olshanski. The theory of these functions arose from the asymptotic theory of Vershik and Kerov of the representations of the symmetric groups.

scholarly journals Character correspondences for symmetric groups and wreath products

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Anton Evseev
Keyword(s):  
Finite Group ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Irreducible Characters ◽  
Arbitrary Finite Group

AbstractThe Alperin–McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its

scholarly journals Young–Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants

2020 ◽  
pp. 2150123
Author(s):  
A. Brini ◽  
A. Teolis
Keyword(s):  
Polynomial Algebra ◽  
Symmetric Groups ◽  
Lie Superalgebras ◽  
General Linear ◽  
Linear Combinations ◽  
Irreducible Characters ◽  
New Class ◽  
Linear Generators ◽  
Capelli Identities ◽  
Quantum Immanants

The set of standard Capelli bitableaux and the set of standard Young–Capelli bitableaux are bases of [Formula: see text], whose action on the Gordan–Capelli basis of polynomial algebra [Formula: see text] have remarkable properties (see, e.g. [A. Brini, A. Palareti and A. Teolis, Gordan–Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988) 1330–1333; A. Brini and A. Teolis, Young–Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989) 775–778; A. Brini and A. Teolis, Capelli bitableaux and [Formula: see text]-forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990) 56–60; A. Brini and A. Teolis, Capelli’s theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA  90 (1993) 10245–10249.]). We introduce a new class of elements of [Formula: see text], called the Capelli immanants, that can be efficiently computed and provide a system of linear generators of [Formula: see text]. The Okounkov quantum immanants [A. Okounkov, Quantum immanants and higher Capelli identities, Transform Groups 1 (1996) 99–126; A. Okounkov, Young basis, Wick formula, and higher Capelli identities, Int. Math. Res. Not. 1996(17) (1996) 817–839.] — quantum immanants, for short — are proved to be simple linear combinations of diagonal Capelli immanants, with explicit coefficients. Quantum immanants can also be expressed as sums of double Young–Capelli bitableaux. Since double Young–Capelli bitableaux uniquely expands into linear combinations of standard Young–Capelli bitableaux, this leads to canonical presentations of quantum immanants, and, furthermore, it does not involve the computation of the irreducible characters of symmetric groups.

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