49 research outputs found
New fermionic formula for unrestricted Kostka polynomials
A new fermionic formula for the unrestricted Kostka polynomials of type
is presented. This formula is different from the one given by
Hatayama et al. and is valid for all crystal paths based on
Kirillov-Reshetihkin modules, not just for the symmetric and anti-symmetric
case. The fermionic formula can be interpreted in terms of a new set of
unrestricted rigged configurations. For the proof a statistics preserving
bijection from this new set of unrestricted rigged configurations to the set of
unrestricted crystal paths is given which generalizes a bijection of Kirillov
and Reshetikhin.Comment: 35 pages; reference adde
Crystal structure on rigged configurations
Rigged configurations are combinatorial objects originating from the Bethe
Ansatz, that label highest weight crystal elements. In this paper a new
unrestricted set of rigged configurations is introduced for types ADE by
constructing a crystal structure on the set of rigged configurations. In type A
an explicit characterization of unrestricted rigged configurations is provided
which leads to a new fermionic formula for unrestricted Kostka polynomials or
q-supernomial coefficients. The affine crystal structure for type A is obtained
as well.Comment: 20 pages, 1 figure, axodraw and youngtab style file necessar
Inhomogeneous lattice paths, generalized Kostka polynomials and A supernomials
Inhomogeneous lattice paths are introduced as ordered sequences of
rectangular Young tableaux thereby generalizing recent work on the Kostka
polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon.
Motivated by these works and by Kashiwara's theory of crystal bases we define a
statistic on paths yielding two novel classes of polynomials. One of these
provides a generalization of the Kostka polynomials while the other, which we
name the A supernomial, is a -deformation of the expansion
coefficients of products of Schur polynomials. Many well-known results for
Kostka polynomials are extended leading to representations of our polynomials
in terms of a charge statistic on Littlewood-Richardson tableaux and in terms
of fermionic configuration sums. Several identities for the generalized Kostka
polynomials and the A supernomials are proven or conjectured. Finally,
a connection between the supernomials and Bailey's lemma is made.Comment: 44 pages, figures, AMS-latex; improved version to appear in Commun.
Math. Phys., references added, some statements clarified, relation to other
work specifie
Character Formulae of -Modules and Inhomogeneous Paths
Let B_{(l)} be the perfect crystal for the l-symmetric tensor representation
of the quantum affine algebra U'_q(\hat{sl(n)}). For a partition mu =
(mu_1,...,mu_m), elements of the tensor product B_{(mu_1)} \otimes ... \otimes
B_{(mu_m)} can be regarded as inhomogeneous paths. We establish a bijection
between a certain large mu limit of this crystal and the crystal of an
(generally reducible) integrable U_q(\hat{sl(n)})-module, which forms a large
family depending on the inhomogeneity of mu kept in the limit. For the
associated one dimensional sums, relations with the Kostka-Foulkes polynomials
are clarified, and new fermionic formulae are presented. By combining their
limits with the bijection, we prove or conjecture several formulae for the
string functions, branching functions, coset branching functions and spinon
character formula of both vertex and RSOS types.Comment: 42 pages, LaTeX2.0
An Invitation to the Generalized Saturation Conjecture
We report about some results, interesting examples, problems and conjectures
revolving around the parabolic Kostant partition functions, the parabolic
Kostka polynomials and ``saturation'' properties of several generalizations of
the Littlewood--Richardson numbers.Comment: 79 pages, new sections, new results and example