A fundamental problem in representation theory is to find an explicit positive rule, akin to the Littlewood-Richardson rule, for decomposing the tensor product of two irreducible representations ofâ€¦ (More)

We investigate Hadwigerâ€™s conjecture for graphs with no stable set of size 3. Such a graph on at least 2t âˆ’ 1 vertices is not tâˆ’ 1 colorable, so is conjectured to have a Kt minor. There is aâ€¦ (More)

Describing minimal generating sets of toric ideals is a well-studied and difficult problem. Neil White conjectured in 1980 that the toric ideal associated to a matroid is generated by quadricsâ€¦ (More)

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent Ï‰ of matrix multiplication by reducing matrix multiplication to group algebra multiplication. In 2005 Cohn,â€¦ (More)

A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product ofâ€¦ (More)

Motivated by our recent work relating canonical bases to combinatorics of Garsia-Procesi modules [1], we give an insertion algorithm that computes the catabolizability of the insertion tableau of aâ€¦ (More)

We identify a subalgebra Ä¤ + n of the extended affine Hecke algebra Ä¤n of type A. The subalgebra Ä¤ + n is a u-analogue of the monoid algebra of Snâ‹‰Z n â‰¥0 and inherits a canonical basis from that ofâ€¦ (More)

The nonstandard Hecke algebra Èžr was defined in [11] to study the Kronecker problem. We study a quotient Èžr,2 of Èžr , called the nonstandard Temperley-Lieb algebra, which is simpler than Èžr. It isâ€¦ (More)

The lowest two-sided cell of the extended affine Weyl group We is the set {w âˆˆ We : w = x Â· w0 Â· z, for some x, z âˆˆ We}, denoted W(Î½). We prove that for any w âˆˆ W(Î½), the canonical basis element C wâ€¦ (More)

We further develop the theory of inducing W -graphs worked out by Howlett and Yin in [6], [7], focusing on the case W = Sn. Our main application is to give two W -graph versions of tensoring with theâ€¦ (More)