8,012 research outputs found
Comultiplication rules for the double Schur functions and Cauchy identities
The double Schur functions form a distinguished basis of the ring
\Lambda(x||a) which is a multiparameter generalization of the ring of symmetric
functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended
to \Lambda(x||a) in a natural way so that the double power sums symmetric
functions are primitive elements. We calculate the dual Littlewood-Richardson
coefficients in two different ways thus providing comultiplication rules for
the double Schur functions. We also prove multiparameter analogues of the
Cauchy identity. A new family of Schur type functions plays the role of a dual
object in the identities. We describe some properties of these dual Schur
functions including a combinatorial presentation and an expansion formula in
terms of the ordinary Schur functions. The dual Littlewood-Richardson
coefficients provide a multiplication rule for the dual Schur functions.Comment: 44 pages, some corrections are made in sections 2.3 and 5.
GENERATING FUNCTIONS FOR REAL CHARACTER DEGREE SUMS OF FINITE GENERAL LINEAR AND UNITARY GROUPS
Abstract. We compute generating functions for the sum of the realvalued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius-Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall-Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius-Schur indicator 1 or −1. 201
GENERATING FUNCTIONS FOR REAL CHARACTER DEGREE SUMS OF FINITE GENERAL LINEAR AND UNITARY GROUPS
Abstract. We compute generating functions for the sum of the realvalued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius-Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall-Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius-Schur indicator 1 or −1. 201
Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
We show that the geometric lifting of the RSK correspondence introduced by
A.N. Kirillov (2001) is volume preserving with respect to a natural product
measure on its domain, and that the integrand in Givental's integral formula
for GL(n,R)-Whittaker functions arises naturally in this context. Apart from
providing further evidence that Whittaker functions are the natural analogue of
Schur polynomials in this setting, our results also provide a new
`combinatorial' framework for the study of random polymers. When the input
matrix consists of random inverse gamma distributed weights, the probability
distribution of a polymer partition function constructed from these weights can
be written down explicitly in terms of Whittaker functions. Next we restrict
the geometric RSK mapping to symmetric matrices and show that the volume
preserving property continues to hold. We determine the probability law of the
polymer partition function with inverse gamma weights that are constrained to
be symmetric about the main diagonal, with an additional factor on the main
diagonal. The third combinatorial mapping studied is a variant of the geometric
RSK mapping for triangular arrays, which is again showed to be volume
preserving. This leads to a formula for the probability distribution of a
polymer model whose paths are constrained to stay below the diagonal. We also
show that the analogues of the Cauchy-Littlewood identity in the setting of
this paper are equivalent to a collection of Whittaker integral identities
conjectured by Bump (1989) and Bump and Friedberg (1990) and proved by Stade
(2001, 2002). Our approach leads to new `combinatorial' proofs and
generalizations of these identities, with some restrictions on the parameters.Comment: v2: significantly extended versio
Bijective proofs for Schur function identities which imply Dodgson's condensation formula and Pl\"ucker relations
We present a ``method'' for bijective proofs for determinant identities,
which is based on translating determinants to Schur functions by the
Jacobi--Trudi identity. We illustrate this ``method'' by generalizing a
bijective construction (which was first used by Goulden) to a class of Schur
function identities, from which we shall obtain bijective proofs for Dodgson's
condensation formula, Pl\"ucker relations and a recent identity of the second
author.Comment: Co-author Michael Kleber added a new proof of his theorem by
inclusion-exclusio
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