Crystals and Their Generalizations

1. Let \(\mu \in P\) be an integral weight, and \(\lambda \in P_+\) a dominant integral weight (such that “\(\lambda \gg \mu\)”). Let \(w\) be an element of the finite Weyl group \(W_{fin}\). Denote by \(V_w^-(\lambda+\mu)\) the (opposite) Demazure module of lowest weight \(w(\lambda+\mu)\) in the level-zero extremal weight module \(V(\lambda+\mu)\). I'd like to explain a Chavalley type formula for the expansion of the graded character of \(V_w^-(\lambda+\mu)\) as a \(\mathbb{Z}[P][[q^{-1}]]\)-linear combination of the graded characters of \(V_x^-(\lambda), x \in W_{af}\), where \(W_{af}\) is the affine Weyl group.
2. I'd like to explain a Monk type formula for the expansion of the graded character of \(V_w^-(\lambda)\) multiplied by \(e^{\mu}\) as a \(\mathbb{Z}[q]\)-linear combination of the graded characters of \(V_x^-(\nu)\), \(x \in W_{af}\) and \(\nu \in P\).

Nakashima and Zelevinsky invented ‘polyhedral realization’, which is a kind of description of crystal bases as lattice points in some polyhedral convex cone.

To construct the polyhedral realization, we need an infinite sequence \(\iota\) of indices. In the case \(\iota\) satisfies a ‘positivity condition’, Nakashima and Zelevinsky also found a method to obtain an explicit form of the polyhedral realization associated with \(\iota\). However, it seems to be difficult to check whether \(\iota\) satisfies the positivity condition or not.

In this talk, I will give a sufficient condition of \(\iota\) for the positivity condition and an explicit form of the polyhedral realization in the case \(\iota\) satisfies the sufficient condition. This is a joint work with Toshiki Nakashima.

The theory of Mirković–Vilonen (MV) polytopes provides a beautiful combinatorial model of crystals that connects various disparate constructions in representation theory. By work of many people, we have explicit isomorphisms between the crystal structures on MV polytopes and on (1) Mirković–Vilonen cycles, (2) PBW bases in quantum groups, (3) preprojective varieties, and (4) KLR modules. In the past ten years, a flurry of activity has birthed a theory of affine MV polytopes. In particular, the connections between affine MV polytopes and the affine versions of (2), (3), and (4) have been established.

I will review the current state of affairs, and I will present my work with Peter Tingley that establishes connections (2) and (3) in the affine case. However, connection (1) between affine MV polytopes and affine MV cycles remains frustratingly open. I will discuss my current work that establishes the connection between affine MV cycles and “undecorated” affine MV polytopes in type A. The argument builds on my earlier work that established an isomorphism between type A affine MV cycles and the Naito–Sagaki–Saito crystal.

Coxeter introduced Coxeter groups in 1934, and he classified the finite Coxeter groups in 1935. Those are classified into \({\mathrm{A}}_n\), \({\mathrm{B}}_n = {\mathrm{C}}_n\), \({\mathrm{D}}_n\), \({\mathrm{F}}_4\), \({\mathrm{E}}_6\), \({\mathrm{E}}_7\), \({\mathrm{E}}_8\), \({\mathrm{G}}_2\), \({\mathrm{I}}_n\), \({\mathrm{H}}_3\) and \({\mathrm{H}}_4\). Coxeter groups appear in many areas of Algebra and Geometry. One of the areas is the representation theory of Lie algebras. Perhaps from 1970's, many researchers have considered that they needed ‘Coxeter groupoids’ being applied for study of the representation theory of Lie superalgebras. In 2000's, ‘Coxeter groupoids’ also became necessary for study of Hopf algebras called ‘Generalized quantum groups (GQGs)’. A GQG can be a quantum group, a multi-parameter quantum group, a quantum group at a root of unity, a quantum superalgebra, or the quantum double of a Nichols algebra of diagonal-type.

In this talk, we mainly explain Bruhat order of Coxeter groupoids, Kharchenko PBW theorem and some important equations in the positive part of the \(A^{(1)}_1\) type affine quantum group.