Joint Algebra Seminar

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A bijection between evil-avoiding and rectangular permutations

The asymmetric simple exclusion process (ASEP), a type of interacting particle system, has been a topic of recent interest in mathematics and physics, with H.T. Yau describing it as a "default stochastic model for transport phenomena." A kind of ASEP, the inhomogeneous TASEP, has unexpected connections to Schubert polynomials and evil-avoiding permutations, the latter of which was introduced by Kim and Williams in 2022. A related class of permutations, called rectangular permutations, was introduced by Chirivì, Fang, and Fourier in 2021, and arises in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evil-avoiding and rectangular permutations in Sn that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the 1-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length 2n-2 in a path of seven vertices starting and ending at the middle vertex.

Minimum polyhedron of n vertices

The unit volume convex body having minimum surface area is a ball. We consider a discrete version of this problem, i.e., the convex polyhedron of n vertices having minimum surface area. F.~Toth claimed that all faces of the minimum polyhedron are triangles. His argument is very intriguing but there was an error. I will explain his beautiful idea and then show how we save the proof. Then I shall give possible minimum shapes for n less than 13.

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Tensor decomposition for a coideal subalgebra and type B and D Jones-Wenzl projectors

One of the most elementary yet beautiful cases of Schur-Weyl duality is that between U_q(gl2) and the Hecke-algebra of the symmetric group acting on tensor powers of the standard representation. Here the Hecke algebra takes a new form as a quotient known as the Temperley-Lieb algebra. This algebra has been well-studied and understood in terms of special idempotents known as the Jones-Wenzl projectors with a diagrammatic meaning. I will explain how to enhance the story from the symmetric group/Temperley-Lieb side to a type B/D version, by replacing U_q(gl2) with a certain coideal subalgebra. I present the type B/D Jones-Wenzl projectors one discovers on the way and many other generalizations from type A to type B/D including convergent powers of type D full-twists. This is based joint work with Catharina Stroppel.

Juggling patterns, Grassmann necklaces, and affine flag varieties

Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. These quiver Grassmannians admit explicit embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.

From Braids to Transverse Slices in Reductive Groups

Slodowy slices and their quantisations play a central role in the geometric Langlands pogramme. We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties. Also building upon recent work of He-Nie, our perspective furnishes a common generalisation and a simple geometric criterion for Weyl group elements to yield strictly transverse slices. We finish by briefly discussing their Poisson structures and quantisations.

Characters and character sheaves of finite groups of Lie type

An important task in the representation theory of finite groups is the determination of their character tables. As the classification of finite simple groups shows, the main difficulties in this context concern the finite groups of Lie type, which arise as an infinite series of finite groups associated to a certain algebraic group over a field of positive characteristic. In order to generically tackle the problem of determining the character tables of finite groups of Lie type, Lusztig developed the theory of character sheaves in the 1980s. In this framework, due to the work of Lusztig and Shoji, the problem is in principle reduced to determining certain roots of unity. We report on some recent progress in this area.

Multiprojective Seshadri stratifications for flag varieties

In 2021 Rocco Chirivi, Xin Fang and Peter Littelmann introduced the notion of a Seshadri stratification on an embedded projective variety, which (among many other interesting consequences) gives rise to a standard monomial theory on the homogeneous coordinate ring, at least in certain nice cases. We will generalize Seshadri stratifications to the multiprojective setting and give a geometric construction of the usual standard monomial theory for flag varieties in types A/B/C/F/G.

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