4,056 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
The pillowcase distribution and near-involutions
In the context of the Eskin-Okounkov approach to the calculation of the
volumes of the different strata of the moduli space of quadratic differentials,
the important ingredients are the pillowcase weight probability distribution on
the space of Young diagrams, and the asymptotic study of characters of
permutations that near-involutions. In this paper we present various new
results for these objects. Our results give light to unforeseen difficulties in
the general solution to the problem, and they simplify some of the previous
proofs.Comment: This paper elaborates on some of the results of the author's PhD
thesis (arXiv:1209.4333). This is the published version,
http://ejp.ejpecp.org/article/view/362
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
Series of nilpotent orbits
We organize the nilpotent orbits in the exceptional complex Lie algebras into
series using the triality model and show that within each series the dimension
of the orbit is a linear function of the natural parameter a=1,2,4,8,
respectively for f_4,e_6,e_7,e_8. We also obtain explicit representatives in a
uniform manner. We observe similar regularities for the centralizers of
nilpotent elements in a series and graded components in the associated grading
of the ambient Lie algebra. More strikingly, for a greater than one, the
degrees of the unipotent characters of the corresponding Chevalley groups,
associated to these series through the Springer correspondance are given by
polynomials which have uniform expressions in terms of a.Comment: 20 pages, revised version with more formulas for unipotent character
A bideterminant basis for a reductive monoid
We use the rational tableaux introduced by Stembridge to give a bideterminant
basis for a normal reductive monoid and for its variety of noninvertible
elements. We also obtain a bideterminant basis for the full coordinate ring of
the general linear group and for all its truncations with respect to saturated
sets. Finally, we deduce an alternative proof of the double centraliser theorem
for the rational Schur algebra and the walled Brauer algebra over an arbitrary
infinite base field which was first obtained by Dipper, Doty and Stoll
- …