New York Applied Algebra Colloquium
Time: Fridays at 11:00 AM - 12:00 PM
Place: CUNY Graduate Center, Room 3209
365 Fifth Avenue (between 34 and 35th streets), New York, NY.
Fall 2016 Schedule
| September|| |
| 9||Speaker: Itamar Stein (Bar-Ilan University)|
Title: The representation theory of the monoid of all partial functions on a finite set as an EI-category algebra.
Abstract: Given a finite monoid M and a field K, one can consider the monoid algebra KM. Unlike group algebras, monoid algebras are seldom semisimple, even if K has a good characteristic. One of the main goals of the study of monoid representations is to understand important invariants of the monoid algebra KM. These include the ordinary quiver, quiver presentation, global dimension, etc - all of which are trivial in the semisimple case hence does not appear in ordinary group representation theory.
In this talk we will discuss the monoid of all partial functions on an n-element set, denoted PT_n, which is one the most important and well-studied finite monoids. We will give an explicit description of the quiver of PT_n where K=C is the field of complex numbers.
Our description uses an isomorphism between the algebra of PT_n and the algebra of the category of all epimorphisms between subsets of an n-element set. This is an EI-category, which means that each endomorphism in the category is an isomorphism. This isomorphism of algebras is an extension of a well known isomorphism discovered by Ben Steinberg.
To describe the quiver of this category we use results of Stuart Margolis, Ben Steinberg and Liping Li on the quiver of finite EI-categories which enables us to reduce to a combinatorial question in the representation theory of symmetric groups. The explicit description we will obtain is in terms of Young diagrams and branching rules of the symmetric group.
| 23||Speaker: Sam van Gool (City College of New York, CUNY)|
Title: Proaperiodic monoids and model theory
Abstract: We begin with the observation that the free profinite aperiodic monoid over a finite set A is isomorphic to the Stone dual space (spectrum) of the Boolean algebra of first-order definable sets of finite A-labelled linear orders (“A-words”). This means that elements of this monoid can be viewed as elementary equivalence classes of models of the first-order theory of finite A-words. From this perspective, the operations of multiplication and ω-power on proaperiodic monoids can be understood in a very concrete way. This point of view allows us to import methods from both topology and model theory, in particular saturated models, into the study of proaperiodic monoids. We use these methods to prove results about ω-terms in the free proaperiodic monoid and well-quasi-orders of factors in related proaperiodic monoids.