Related papers: On the representation theory of finite J-trivial m…
The usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously…
For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The…
Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation,…
We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke…
We show that a proper degeneracy at $q=0$ of the $q$-deformed rook monoid of Solomon is the algebra of a monoid $R_n^0$ namely the $0$-rook monoid, in the same vein as Norton's $0$-Hecke algebra being the algebra of a monoid $H_n^0 =…
A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob--Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic…
A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, i.e. subvarieties of the varieties of representations. The study of this monoid leads to interesting…
We investigate the representation theory of certain specializations of the Ariki-Koike algebras, obtained by setting $q=0$ in a suitably normalized version of Shoji's presentation. We classify the simple and projective modules, and describe…
This paper develops the fundamentals of modular representation theory for finite monoids, introducing the decomposition matrix and exploring its connection to Brauer characters. We define modular characteristic and explain how the…
After reformulating the representation theory of 0-Hecke algebras in an appropriate family of Yang-Baxter bases, we investigate certain specializations of the Ariki-Koike algebras, obtained by setting q=0 in a suitably normalized version of…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
In a previous paper we prove that any semisimple triangular Hopf algebra A over an algebraically closed field of characteristic 0 (say the field of complex numbers C) is obtained from a finite group after twisting the ordinary…
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify…
Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is…
Solitons in two-dimensional quantum field theory exhibit patterns of degeneracies and associated selection rules on scattering amplitudes. We develop a representation theory that captures these intriguing features of solitons. This…
We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term…
The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a…
We develop a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the…
These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in…
The (ordinary) quiver of an algebra $A$ is a graph that contains information about the algebra's representations. We give a description of the quiver of $\mathbb{C}PT_{n}$, the algebra of the monoid of all partial functions on $n$ elements.…