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SCIENTIFIC PROGRAMS AND ACTIVITIES |
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| December 23, 2024 |
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A Talk (Walk)on
the McKay Correspondence
by Georgia Benkart, University of Wisconsin-Madison The McKay correspondence establishes deep connections between finite
subgroups of SU(2) and affine simply-laced Lie algebras. This talk
will relate those objects to certain associative algebras, their
combinatorics and representations, and to walks on Dynkin diagrams. Generalization of nil-coxeter monoids In algebraic combinatorics, the nil-coxeter monoid encodes many
interesting geometric and representation theoretic properties related
to the symmetric groups.Here I will present generalizations of nil-coxeter
monoid that plays similar roles related to schubert carieties and
the affine coxeter groups of type A. Algebraic monoids and equivariant embeddings of algebraic groups This mini-course will discuss structure results and problems for
algebraic monoids (possibly non-linear) and their relation to equivariant
embeddings of algebraic groups. In particular, we shall present
a result of Rittatore: any irreducible algebraic monoid with linear
unit group is linear. Also, we shall show that every irreducible
algebraic monoid is obtained from a linear one by a process of ïnduction". Affine Permutations and an Affine Coxeter Monoid We describe a affine version of the monoid of non-decreasing parking
functions, which may be obtained as a quotient of the zero-Hecke
algebra of the affine symmetric group, given combinatorially by
a set of generalized Dyck paths. We will also describe connections
between the affine Coxeter monoid and the study of 321-avoiding
affine permutations. Representation theory of reductive normal algebraic monoids The representation theory of a reductive normal algebraic monoid forms an interesting part of that of its algebraic group of units. The representation category of the monoid in question splits into a direct sum of "highest weight" subcategories (in the sense of Cline-Parshall-Scott), each of which is controlled by a finite dimensional quasihereditary algebra. These quasihereditary algebras are natural examples of generalized Schur algebras, introduced and studied by S. Donkin in the 1980s.
The kernel and Unipotent Radicals in Linear Algebraic Monoids Let M be an irreducible linear algebraic monooid, G its unit group,
ker(M) its minimal (two-sided) semigroup-theoretical ideal and R_u(G)
the unipotent radical of G. There is close structure connection
between ker(M) and R_u(G). Some relevant problems about decompositions
of R_u(G) and regularity of M are also discussed. Groups, Semigroups, Semigroup Rings and Set-Theoretic Solutions
of the Yang-Baxter Equation In this minicourse we introduce several algebraic structures that appear natural in the search for set-theoreric solutions of the Yang-Baxter equation. We focus on non-degenerate involutive solutions. The algebraic structures include monoids and groups of I-type, involutive Yang-Baxter groups, braces and semigroup rings. The main properties of these structures will be discussed as well as their relevance for finding set-theoretic solutions. Several open problems and some solutions will be presented. -------------------------------------------------------------------------------- On the global dimension of left regular band algebras and Leray
numbers It has been realized in recent years, that many combinatorial structures have the structure of a left regular band. The representation theory of left regular bands has shed light on these objects as well as aid in analyzing Markov chains on them, with surprising and beautiful results. We begin by recalling definitions and properties of left regular bands and related constructions. We then survey several examples of left regular bands that have been extensively studied over the past 15 years. These include left regular bands associated to hyperplane arrangements, ordered matroids and interval greedoids. We illustrate how some of the left regular bands that have appeared in the literature are special cases of semigroup-theoretic constructions. We also introduce some new examples: the free partially commutative left regular bands, which generalizes trace monoids and right angled Artin groups; geometric left regular bands, which includes all the left regular bands that have appeared in the algebraic combinatorics literature; and the left regular band of an acyclic quiver whose semigroup algebra is equal to the path algebra of the quiver. We begin with a new description of the quiver of a left regular band algebra. We show that the n-th Ext space between simple modules is the reduced n-1 cohomology of an associated interval in the order complex of Green's right order of the monoid. This shows that the global dimension is bounded above by the Leray number of the order complex of its Green right order. This leads to characterisation of the free partially commutative left regular bands with a hereditary algebra as those constructed from chordal graphs. Our main tools are classifying spaces and the cohomology of monoids and small categories. Although we are mostly interested in monoid cohomology, which is a natural generalization of group cohomology, we will also need to work with categories that are not monoids; namely, posets and the semidirect product of a monoid with a set. This is a joint work with Franco Saliola, Departement de Mathematiques LaCIM, Universite du Quebec a Montreal, and Benjamin Steinberg, Department of Mathematics, CCNY, CUNY
Conjugacy classes of left ideals of finite dimensional algebras For a finite dimensional unital algebra A over a field K let C(A) denote the set of conjugacy classes of left ideals in A. Then C(A) can be considered as a semigroup under the natural operation induced from the multiplication in A. The motivating idea is to look for invariants of A that can be expressed in terms of C(A). Two of the main problems can be formulated as follows: Problem 1 Determine necessary and sufficient conditions under which C(A) is finite. Problem 2 Determine properties of the algebra A that can be recognized by the semigroup C(A). If K is algebraically closed, then A can be considered as a connected algebraic monoid. The definition of C(A) was introduced in a joint paper with L.Renner (2003). The finiteness of C(A) is strongly related to the finite representation type property of finite dimensional algebras. Another motivation for a study of C(A) follows from its relation to the double cosets U(A)aU(A), where U(A) is the group of units of A, for a ? A. In particular, it is known that if C(A) is finite then the number of such cosets also is finite. This is a joint work with A. M ecel.
Regular linear algebraic semigroups Reductive linear algebraic monoids M with 0 have been studied for
over 30 years. Equivalently M is regular (a ? aMa for all a ? M).
What happens when M is not assumed to have an identity element?
Here we study irreducible regular linear algebraic semigroups S.
Such semigroups have a maximal J-class J. We focus on the situation
when J/R and J/L are projective varieties. This is a natural assumption.
For instance for the J-class J of rank r matrices, J/R and J/L are
Grassmanian spaces. In general such semigroups S arise naturally
within a reductive monoid. We explore the abstract structure of
S, from local (Green-Rees) to global (Rhodes). The Betti Numbers of Rationally Smooth Simple Embeddings Let G be a simple algebraic group with Weyl group (W, S) and let J ? S.We identify a certain subset SJ ? WJ that plays the role of S ? W in the familiar case J = Ø. We end up with the descent system (WJ, SJ), which records important combinatorial information about the cellular structure of certain group embeddings. Also associated with any proper subset J ? S is the simple G×G-embedding P(J). We identify those subsets J of S with the desirable property that P(J) is rationally smooth as an algebraic variety. In such cases P(J) decomposes into a union of "rational cells" via a "monoid BB-decomposition". The descent system is the combinatorial device that is needed to help quantify the Betti numbers of P(J) as well as those of a certain closely related torus embedding X(J) ? P(J). We illustrate the theory with many examples. The Betti numbers of X(J) are, in many cases, related to Eulerian polynomials through the work of Procesi and Golubitzky. The Betti numbers of the wonderful embedding (a particular case of J = Ø) were originally obtained by De Concini and Procesi. We explain how their results can be seen as a motivation for introducting descent systems.
The endomorphisms monoid of a homogeneous vector bundle Since the early works of Putcha and Renner in the '80, affine algebraic monoids have been thoroughly studied by several authors. Nowadays, the relationship between the geometry and the algebraic structure of such monoids, their systems of idempotents, their representation theory... are reasonably understood. However, the study of arbitrary algebraic monoids is not developed in such extent. In 2007 an initial step was taken in this direction, with the generalization of Chevalley's decomposition of algebraic groups to the case of normal algebraic monoids (Brion, ). This structure theorem suggests that a representation theory of algebraic monoids should be developed in the context of homogeneous vector bundles over abelian varieties. In this context, the endomorphisms monoid of such bundles should play the role of End(\bfkn), the monoid of n×n-matrices, in the case of affine algebraic monoids. In this talk be will describe the geometric (as vector bundles) and algebraic (as monoids)structure of these endomorphism monoids. This is a joint work with L. Brambila-Paz.
A 0-Hecke monoid action on reduced words Consider the set of reduced words of a given Coxeter element in the alphabet of all reflections, equivalently, the set of all maximal chains in the non-crossing partition lattice . This set is endowed with a graph structure, by connecting two chains if they differ in exactly one element. A 0-Hecke monoid action on the vertex set is introduced. This allows us to define a well-behaved natural weak order on this set, implying an evaluation of the radius of this graph.
Monoids in algebraic combinatorics This minicourse focuses on the appearance and applications of monoids in algebraic combinatorics. The first example we will discuss is the Tsetlin library. Suppose that a library consists of n books b1, ..., bn on a single shelf. Assume that only one book is picked at a time and is returned before the next book is picked up. The book bi is picked with probability xi and placed at the end of the shelf. This gives rise to a famous Markov chain, which was studied by many people. We will then discover that it is part of a much larger family of Markov chains defined by a generalization of Schützenberger's promotion operator on the set L of linear extensions of a finite poset of size n. The Tsetlin library corresponds to the case of the anti-chain. Studying the eigenvalues and multiplicities of the transition matrix, we will find interesting seemingly new R-trivial monoids. Using Steinberg's extension of Brown's theory for Markov chains on left regular bands, we can derive all results. In the process of discovering and deriving the above results, we will also make use of computational tools in the open source mathematical software system Sage. The second main example we will explore relates to Fomin and Greene's
theory of noncommutative symmetric functions and its affine analogue.
In particular, we will investigate noncommutative symmetric functions
in terms of the affine nilCoxeter algebra as introduced by Lam and
the affine local plactic algebra as used by Korff and Stroppel.
These provide tools to study structure coefficients, in particular
generalizations of Littlewood-Richardson coefficients. Direct product factorization of bipartite graphs with partition-reversing
involutions Bresar, Imrich, Klavzar, Rall and Zmazek partially answered the
question of finding all graphs X such that K2 ×X ? Qn, where
× is the direct product of graphs, Qn the n-dimensional hypercube
and K2 the complete graph on two vertices. In this talk we completely
resolve the question by solving K2 ×X ? B, where B is bipartite.
We show that the solutions X are in one-to-one correspondence with
the Aut(B)-conjugacy classes of the set of partition-reversing involutions
of B. Conjugacy Order in Canonical and Dual Canonical Monoids Reductive monoids are Zariski closures of reductive groups. A reductive monoid, say M, decomposes nicely in terms of its unit group G and a set of idempotents L, called the cross-section lattice, that indexes the G×G-orbits of M. A canonical monoid is a reductive monoid whose cross-section lattice has a unique minimal non-zero element - that is, it has a minimal non-zero G×G-orbit - while a dual canonical monoid is a reductive monoid whose cross-section lattice is the dual of the cross-section lattice of a canonical monoid. In the literature, canonical and dual canonical monoids are called J-irreducible and J-coirreducible monoids, respectively, both of type Ø. In this talk I will describe the partial order that arises in a
decomposition of a reductive monoid, in terms of conjugacy classes.
In particular, I will discuss some results describing the partial
order on this poset, both within and between classes indexed by
the cross-section lattice, in the cases of both canonical and dual
canonical monoids. The description between classes is modeled on
Putcha's projection maps between J-classes in reductive monoids. Algorithmic representation theory of finite monoids In the past decades, a number of algebras were studied for their representation theory in algebraic combinatorics, and many of them turned out to be finite monoid algebras. These often arise in the context of Coxeter groups and are endowed with a strong structure, which gives a chance to describe explicitly some pieces of their representation theory. For example, the biHecke monoids are one of the very first non-basic aperiodic monoid for which there is a complete combinatorial description of the simple modules. The study of such monoids, based on heavy computer exploration, prompted the development of algorithms and their implementation, and shed some light on the general theory. In this talk, we review some algorithms for the representation theory of monoids algebras. In particular, we describe how the Cartan invariant matrix can be expressed, in characteristic zero, using characters and some simple combinatorial statistic. We deduce an algorithm to compute it efficiently from the composition factors of the left and right class modules. For example, the computation took one hour for a monoid of size 31103 with 120 simple modules. When the monoid is aperiodic, this approach works in any characteristic, and generalizes to a principal ideal domain like Z. When the monoid is J-trivial, we retrieve the formerly known purely combinatorial description of the Cartan matrix. The talk will be illustrated by concrete computations using the
open source mathematical software system Sage. On Linear Hodge-Newton Decomposition for Reductive Monoids Recently R. Kottwitz and E. Viehmann, in course of their study of certain affine Springer fibers, developed linear variants of Katz's Hodge-Newton decomposition - for an arbitrary reductive group over a p-adic field, as well as for the monoid of n n matrices over a p-adic field. The proof in the general linear monoid case involved perturbing a matrix by a scalar one so as to make it invertible and using the result for the group case. It is thus natural to ask if this phenomenon of linear Hodge-Newton decomposition is monoid-theoretic, for instance whether there is a monoid-theoretic proof. Accordingly, we consider a reductive linear algebraic monoid [`G] over a p-adic field F of characteristic zero, whose group of units is defined and split over the ring of integers of F. We define the notions of "Newton points" and "Hodge points" associated to F-rational points of [`G], and prove a relationship between them, that constitutes part of the linear Hodge-Newton decomposition. Doing so requires one to develop some structure theory for reductive
monoids over p-adic fields. For instance, one needs an Iwasawa decomposition
and a Cartan decomposition for these objects, and a "Bruhat-Tits
inequality" that relates these. A crucial ingredient in proving
the Iwasawa and Cartan decompositions is an adaptation of Renner's
Bruhat decomposition to non-algebraically closed fields. We also
discuss other, related, structure theoretic results such as the
affine Bruhat decomposition, which could be of independent interest. Categorifying Quantum Groups What is categorification? If you de-categorify Vector-Spaces, you replace isomorphism classes of objects with natural numbers (their dimensions), replace direct sum with addition of those numbers, replace tensor product with multiplication. To categorify is to undo this process. For instance, one might start with the ring of symmetric functions and realize it is a shadow of the representation theory of the symmetric group.
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