Rosa C. Orellana

Reduced Kronecker Coefficients and Counter–Examples to Mulmuley’s Strong Saturation Conjecture SH: With an Appendix by Ketan Mulmuley

Abstract.We provide counter–examples to Mulmuley’s strong saturation conjecture (strong SH) for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker coefficient is zero can be done in polynomial time. We also provide a short proof of the #P– hardness of computing the Kronecker coefficients. Both results rely on the connections between the Kronecker coefficients and another family of structural constants in the representation theory of the symmetric groups, Murnaghan’s reduced Kronecker coefficients.An appendix by Mulmuley introduces a relaxed form of the saturation hypothesis SH, still strong enough for the aims of Geometric Complexity Theory.

The partition algebra and the Kronecker coefficients

We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behaviour and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the reduced Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition.

Weights of Markov traces on Hecke algebras

Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

q-centralizer algebras for spin groups

Let V and Vϵ be the vector and a spinor representation of soN, N∈N. We show that the quantum group Uq(soN) and the braid group corresponding to the Dynkin diagram Bf (the latter acting via R-matrices) are each others commutant on Vϵ⊗V⊗f. Moreover, these braid representations factor through specializations of Haring–Oldenburgs B-BMW-algebra; this also holds with Vϵ replaced by Vmϵ, m∈N, if N is even. We use this observation to compute the weights of the Markov trace of this algebra as a 2-variable function, and the values of the parameters for which it is semisimple.

A minimaj-preserving crystal on ordered multiset partitions

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization

The number of permutations with the same peak set for signed permutations

R_{n,k}

A COMBINATORIAL INTERPRETATION FOR THE COEFFICIENTS IN THE KRONECKER PRODUCT s(n p;p) s

due to Haglund, Rhoades and Shimozono of the coinvariant algebra

On the Kronecker Product s (n-p,p) * s λ .

R_n

Graphs with equal chromatic symmetric functions

. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Quasipolynomial formulas for the Kronecker coefficients indexed by two two―row shapes (extended abstract)

A signed permutation \pi = \pi_1\pi_2 \ldots \pi_n in the hyperoctahedral group B_n is a word such that each \pi_i \in {-n, \ldots, -1, 1, \ldots, n} and {|\pi_1|, |\pi_2|, \ldots, |\pi_n|} = {1,2,\ldots,n}. An index i is a peak of \pi if \pi_{i-1} \pi_{i+1} and P_B(\pi) denotes the set of all peaks of \pi. Given any set S, we define P_B(S,n) to be the set of signed permutations \pi \in B_n with P_B(\pi) = S. In this paper we are interested in the cardinality of the set P_B(S,n). In 2012, Billey, Burdzy and Sagan investigated the analogous problem for permutations in the symmetric group, S_n. In this paper we extend their results to the hyperoctahedral group; in particular we show that #P_B(S,n) = p(n)2^{2n-|S|-1} where p(n) is the same polynomial found in by Billey, Burdzy and Sagan which leads to the explicit computation of interesting special cases of the polynomial p(n). In addition we have extended these results to the case where we add \pi_0=0 at the beginning of the permutations, which gives rise to the possibility of a peak at position 1, for both the symmetric and the hyperoctahedral groups.