Induction and restriction of kazhdan-lusztig cells

We interpret the orthogonality relation of Kostka polynomials arising from complex reection groups ([Shoji, Invent. Math. 74 (1983), J. Algebra 245 (2001)] and [Lusztig, Adv. Math. 61 (1986)]) in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counterpart of Kostka polynomials. Then, we show that every generalized Springer correspondence ([Lusztig, Invent. Math. 75 (1984)]) in a good characteristic gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer bers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type BC. The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [Ciubotaru-Kato-K, Invent. Math. 178 (2012)] x3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type A and asymptotic type BC cases, and therefore one can skip geometric sections x3{5 to see the key ideas and basic examples/techniques.

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A homological study of Green polynomials

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

In this paper we will demonstrate that any compact quantum group can be used as symmetry groups for quantum channels, which leads us to the concept of covariant channels. We, then, unearth the structure of the convex set of covariant channels by identifying all extreme points under the assumption of multiplicity-free condition for the associated fusion rule, which provides a wide generalization of some recent results. The presence of quantum group symmetry contrast to the group symmetry will be highlighted in the examples of quantum permutation groups and $SU_q(2)$. In the latter example, we will see the necessity of the Heisenberg picture coming from the non-Kac type condition. This paper ends with the covariance with respect to projective representations, which leads us back to Weyl covariant channels and its fermionic analogue.

Quantum channels with quantum group symmetry

We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space (${\mathbb{Z}}$2≀Sn)/(Sr×Sn−r). Specifically, for r not too far from n/2, we determine that, to first order in n, $\tfrac{1}{4}$n log n steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the “cutoff phenomenon.”

Takeshi Tokuyama

Papers

On the decomposition rules of tensor products of the representations of the classical Weyl groups

A theorem on the representations of the Weyl groups of typeDn andBn − 1