90 research outputs found
A New Multiplicity Formula for the Weyl Modules of Type A
A monomial basis and a filtration of subalgebras for the universal enveloping
algebra of a complex simple Lie algebra of type is given
in this note. In particular, a new multiplicity formula for the Weyl module
of is obtained in this note.Comment: 13 page
Standard monomial theory for wonderful varieties
A general setting for a standard monomial theory on a multiset is introduced
and applied to the Cox ring of a wonderful variety. This gives a degeneration
result of the Cox ring to a multicone over a partial flag variety. Further, we
deduce that the Cox ring has rational singularities.Comment: v3: 20 pages, final version to appear on Algebras and Representation
Theory. The final publication is available at Springer via
http://dx.doi.org/10.1007/s10468-015-9586-z. v2: 20 pages, examples added in
Section 3 and in Section
Metaplectic Ice
Spherical Whittaker functions on the metaplectic n-fold cover of GL(r+1) over
a nonarchimedean local field containing n distinct n-th roots of unity may be
expressed as the partition functions of statistical mechanical systems that are
variants of the six-vertex model. If n=1 then in view of the Casselman-Shalika
formula this fact is related to Tokuyama's deformation of the Weyl character
formula. It is shown that various properties of these Whittaker functions may
be expressed in terms of the commutativity of row transfer matrices for the
system. Potentially these properties (which are already proved by other
methods, but very nontrivial) are amenable to proof by the Yang-Baxter
equation
Standard Monomial Theory for desingularized Richardson varieties in the flag variety GL(n)/B
We consider a desingularization Gamma of a Richardson variety X_w^v=X_w \cap
X^v in the flag variety Fl(n)=GL(n)/B, obtained as a fibre of a projection from
a certain Bott-Samelson variety Z. We then construct a basis of the homogeneous
coordinate ring of Gamma inside Z, indexed by combinatorial objects which we
call w_0-standard tableaux
Crystal constructions in Number Theory
Weyl group multiple Dirichlet series and metaplectic Whittaker functions can
be described in terms of crystal graphs. We present crystals as parameterized
by Littelmann patterns and we give a survey of purely combinatorial
constructions of prime power coefficients of Weyl group multiple Dirichlet
series and metaplectic Whittaker functions using the language of crystal
graphs. We explore how the branching structure of crystals manifests in these
constructions, and how it allows access to some intricate objects in number
theory and related open questions using tools of algebraic combinatorics
Can fusion coefficients be calculated from the depth rule ?
The depth rule is a level truncation of tensor product coefficients expected
to be sufficient for the evaluation of fusion coefficients. We reformulate the
depth rule in a precise way, and show how, in principle, it can be used to
calculate fusion coefficients. However, we argue that the computation of the
depth itself, in terms of which the constraints on tensor product coefficients
is formulated, is problematic. Indeed, the elements of the basis of states
convenient for calculating tensor product coefficients do not have a
well-defined depth! We proceed by showing how one can calculate the depth in an
`approximate' way and derive accurate lower bounds for the minimum level at
which a coupling appears. It turns out that this method yields exact results
for and constitutes an efficient and simple algorithm for
computing fusion coefficients.Comment: 27 page
A New Young Diagrammatic Method For Kronecker Products of O(n) and Sp(2m)
A new simple Young diagrammatic method for Kronecker products of O(n) and
Sp(2m) is proposed based on representation theory of Brauer algebras. A general
procedure for the decomposition of tensor products of representations for O(n)
and Sp(2m) is outlined, which is similar to that for U(n) known as the
Littlewood rules together with trace contractions from a Brauer algebra and
some modification rules given by King.Comment: Latex, 11 pages, no figure
Supersymmetry approach to Wishart correlation matrices: Exact results
We calculate the `one-point function', meaning the marginal probability
density function for any single eigenvalue, of real and complex Wishart
correlation matrices. No explicit expression had been obtained for the real
case so far. We succeed in doing so by using supersymmetry techniques to
express the one-point function of real Wishart correlation matrices as a
twofold integral. The result can be viewed as a resummation of a series of Jack
polynomials in a non-trivial case. We illustrate our formula by numerical
simulations. We also rederive a known expression for the one-point function of
complex Wishart correlation matrices.Comment: 21 pages, 2 figure
- …