104 research outputs found
Symmetric Functions in Noncommuting Variables
Consider the algebra Q> of formal power series in countably
many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...)
of symmetric functions in noncommuting variables consists of all elements
invariant under permutation of the variables and of bounded degree. We develop
a theory of such functions analogous to the ordinary theory of symmetric
functions. In particular, we define analogs of the monomial, power sum,
elementary, complete homogeneous, and Schur symmetric functions as will as
investigating their properties.Comment: 16 pages, Latex, see related papers at
http://www.math.msu.edu/~sagan, to appear in Transactions of the American
Mathematical Societ
Plane overpartitions and cylindric partitions
Generating functions for plane overpartitions are obtained using various
methods such as nonintersecting paths, RSK type algorithms and symmetric
functions. We extend some of the generating functions to cylindric partitions.
Also, we show that plane overpartitions correspond to certain domino tilings
and we give some basic properties of this correspondence.Comment: 42 pages, 11 figures, corrected typos, revised parts, figures
redrawn, results unchange
Symmetric functions in noncommuting variables
Consider the algebra Qhhx1, x2, . . .ii of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x1, x2, . . .) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur
symmetric functions as will as investigating their properties
A combinatorial formula for graded multiplicities in excellent filtrations
A filtration of a representation whose successive quotients are isomorphic to
Demazure modules is called an excellent filtration. In this paper we study
graded multiplicities in excellent filtrations of fusion products for the
current algebra . We give a combinatorial formula for the
polynomials encoding these multiplicities in terms of two dimensional lattice
paths. Corollaries to our main theorem include a combinatorial interpretation
of various objects such as the coeffficients of Ramanujan's fifth order mock
theta functions , Kostka polynomials for hook
partitions and quotients of Chebyshev polynomials. We also get a combinatorial
interpretation of the graded multiplicities in a level one flag of a local Weyl
module associated to the simple Lie algebras of type
Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions
We show that, for a certain class of partitions and an even number of
variables of which half are reciprocals of the other half, Schur polynomials
can be factorized into products of odd and even orthogonal characters. We also
obtain related factorizations involving sums of two Schur polynomials, and
certain odd-sized sets of variables. Our results generalize the factorization
identities proved by Ciucu and Krattenthaler (Advances in combinatorial
mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that
if, in some of the results, the partitions are taken to have rectangular or
double-staircase shapes and all of the variables are set to 1, then
factorization identities for numbers of certain plane partitions, alternating
sign matrices and related combinatorial objects are obtained.Comment: 22 pages; v2: minor changes, published versio
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