6,943 research outputs found
Partitions of the set of nonnegative integers with the same representation functions
For a set of nonnegative integers S let RS(n) denote the number of unordered representations of the integer n as the sum of two different terms from S. In this paper we focus on partitions of the natural numbers into two sets affording identical representation functions. We solve a recent problem of Chen and Lev. © 2017 Elsevier B.V
Combinatorial methods of character enumeration for the unitriangular group
Let \UT_n(q) denote the group of unipotent upper triangular
matrices over a field with elements. The degrees of the complex irreducible
characters of \UT_n(q) are precisely the integers with , and it has been
conjectured that the number of irreducible characters of \UT_n(q) with degree
is a polynomial in with nonnegative integer coefficients (depending
on and ). We confirm this conjecture when and is arbitrary
by a computer calculation. In particular, we describe an algorithm which allows
us to derive explicit bivariate polynomials in and giving the number of
irreducible characters of \UT_n(q) with degree when and . When divided by and written in terms of the variables
and , these functions are actually bivariate polynomials with nonnegative
integer coefficients, suggesting an even stronger conjecture concerning such
character counts. As an application of these calculations, we are able to show
that all irreducible characters of \UT_n(q) with degree are
Kirillov functions. We also discuss some related results concerning the problem
of counting the irreducible constituents of individual supercharacters of
\UT_n(q).Comment: 34 pages, 5 table
A duality between -multiplicities in tensor products and -multiplicities of weights for the root systems or
Starting from Jacobi-Trudi's type determinental expressions for the Schur
functions corresponding to types and we define a natural
-analogue of the multiplicity when is a
tensor product of row or column shaped modules defined by . We prove that
these -multiplicities are equal to certain Kostka-Foulkes polynomials
related to the root systems or . Finally we derive formulas expressing
the associated multiplicities in terms of Kostka numbers
Combinatorial proof for a stability property of plethysm coefficients
Plethysm coefficients are important structural constants in the representation the-
ory of the symmetric groups and general linear groups. Remarkably, some sequences
of plethysm coefficients stabilize (they are ultimately constants). In this paper we
give a new proof of such a stability property, proved by Brion with geometric representation theory techniques. Our new proof is purely combinatorial: we decompose
plethysm coefficients as a alternating sum of terms counting integer points in poly-
topes, and exhibit bijections between these sets of integer points.Ministerio de Ciencia e Innovación MTM2010–19336Junta de AndalucÃa FQM–333Junta de AndalucÃa P12–FQM–269
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