459 research outputs found
Fusion products, Kostka polynomials, and fermionic characters of su(r+1)_k
Using a form factor approach, we define and compute the character of the
fusion product of rectangular representations of \hat{su}(r+1). This character
decomposes into a sum of characters of irreducible representations, but with
q-dependent coefficients. We identify these coefficients as (generalized)
Kostka polynomials. Using this result, we obtain a formula for the characters
of arbitrary integrable highest-weight representations of \hat{su}(r+1) in
terms of the fermionic characters of the rectangular highest weight
representations.Comment: 21 pages; minor changes, typos correcte
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
Galleries and q-analogs in combinatorial representation theory
Schur functions and their q-analogs constitute an interesting branch of combinatorial representation theory. For Schur functions one knows several combinatorial formulas regarding their expansion in terms of monomial symmetric functions, their structure constants and their branching coefficients. In this thesis we prove q-analogs of these formulas for Hall-Littlewood polynomials. We give combinatorial formulas for the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions, for their structure constants and their branching coefficients. Specializing these formulas we get new proofs for the formulas involving Schur functions. As a combinatorial tool we use the gallery model introduced by Gaussent and Littelmann and show its relation to the affine Hecke algebra. All assertions are then proven in the more general context of the Macdonald basis of the spherical Hecke algebra. We show a commutation formula in the affine Hecke algebra with which we obtain a Demazure character formula involving galleries. We give a geometric interpretation of Kostka numbers and Demazure multiplicities of a complex reductive algebraic group using the affine Grassmanian of its Langlands dual group. As a further application we prove some first results regarding the positivity of Kostka-Foulkes coefficients
Branching rules, Kostka-Foulkes polynomials and -multiplicities in tensor product for the root systems and
The Kostka-Foulkes polynomials related to a root system can be
defined as alternated sums running over the Weyl group associated to
By restricting these sums over the elements of the symmetric group when is of type or , we obtain again a class of
Kostka-Foulkes polynomials. When is of type or there exists a
duality beetween these polynomials and some natural -multiplicities in
tensor product \cite{lec}. In this paper we first establish identities for the
which implies in particular that they can be decomposed as sums
of Kostka-Foulkes polynomials related to the root system of type with
nonnegative integer coefficients. Moreover these coefficients are branching
rule coefficients. This allows us to clarify the connection beetween the
-multiplicities and the polynomials defined by Shimozono and Zabrocki in
\cite{SZ}. Finally we establish that the -multiplicities defined for the
tensor powers of the vector representation coincide up to a power of with
the one dimension sum introduced in \cite{Ok} This shows that in this case
the one dimension sums are affine Kazhdan-Lusztig polynomials
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