Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas

We give a complete description of the graded multiplicity space which appears in the Feigin-Loktev fusion product [FL99] of graded Kirillov-Reshetikhin modules for all simple Lie algebras. This construction is used to obtain an upper bound formula for the fusion coefficients in these cases. The formula generalizes the case of g=A_r [AKS06], where the multiplicities are generalized Kostka polynomials [SW99,KS02]. In the case of other Lie algebras, the formula is the the fermionic side of the X=M conjecture [HKO+99]. In the cases where the Kirillov-Reshetikhin conjecture, regarding the decomposition formula for tensor products of KR-modules, has been been proven in its original, restricted form, our result provides a proof of the conjectures of Feigin and Loktev regarding the fusion product multiplicites.Comment: 22 pages; v2: minor changes; v3: exposition clarifie

On the number of partition weights with Kostka multiplicity one

Given a positive integer n, and partitions lambda and mu of n, let K-lambda mu denote the Kostka number, which is the number of semistandard Young tableaux of shape lambda and weight mu. Let J(lambda) denote the number of mu such that K lambda mu = 1. By applying a result of Berenshtein and Zelevinskii, we obtain a formula for J(lambda) in terms of restricted partition functions, which is recursive in the number of distinct part sizes of lambda. We use this to classify all partitions lambda such that J(lambda) = 1 and all lambda such that J(lambda) = 2. We then consider signed tableaux, where a semistandard signed tableau of shape lambda has entries from the ordered set {0 \u3c \u3c 1 \u3c (2) over bar \u3c 2 \u3c ...}, and such that i and (i) over bar contribute equally to the weight. For a weight (omega(0), mu) with mu a partition, the signed Kostka number K-lambda(omega 0, mu)(+/-) is defined as the number of semistandard signed tableaux of shape lambda and weight (omega(0), mu), and J(+/-)(lambda) is then defined to be the number of weights (omega(0), mu) such that K-lambda(omega 0,mu)(+/-) = 1. Using different methods than in the unsigned case, we find that the only nonzero value which J((lambda))(+/-) can take is 1, and we find all sequences of partitions with this property. We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups

A pentagon of identities, graded tensor products and the Kirillov-Reshetikhin conjecture

This paper provides a brief review of the relations between the Feigin-Loktev conjecture on the dimension of graded tensor products of \g[t]-modules, the Kirillov-Reshetikhin conjecture, the combinatorial ``M=N" conjecture, their proofs for all simple Lie algebras, and a pentagon of identities which results from the proof.Comment: 21 page

Some properties of Specht modules for the wreath product of symmetric groups

We investigate a class of modules for the wreath product Sm wr Sn of two symmetric groups which are analogous to the Specht modules of the symmetric group, and prove a range of properties for these modules which demonstrate this analogy. In particular, we prove analogues of the Specht module branching rule, we obtain results on homomorphisms and extensions between these modules, and, over an algebraically closed field whose characteristic is neither 2 nor 3, we prove that, if a module for Sm wr Sn has a filtration by these Specht module analogues, then the multiplicities with which they occur do not depend on the choice of a filtration