AIS - Schur Multiplier and Representation Theory (2025)

Venue: Harish-Chandra Research Institute, Prayagraj

Dates: 4 Dec 2025 to 16 Dec 2025

Convener(s)

Abstract: It is well established that cohomology theory and representation theory of groups  have profound connections. Such connections also exist for category theory, Lie algebras and more.  In the proposed AIS, we have a plan to train students with basic cohomology and representation theory of groups so that they can explore these connections in group theory settings. Starting with the basics of the second cohomology group of a finite group; the restriction, inflation, and transgression homomorphisms will be studied, and the Hochschild-Serre exact sequence for cohomology of groups will be established. We then specialise to the  Schur multiplier M(G) of a finite group G, which is defined to be the second cohomology group of G with coefficients in C^{*}, the group of non-zero complex numbers. The Schur multiplier plays an important role in the theory of extensions of groups and the study of Projective representations of finite groups. Schur Multiplier is also used as a powerful tool in other areas such as algebraic number theory, block theory of group algebras, and classification of finite simple groups etc. On the other hand, representation theory, as is well known, is one of the very fundamental objects in mathematics.  Having its origins in algebra and number theory,  representation theory of groups has applications in diverse areas such as physics, statistics, and engineering, to name a few. This workshop will focus on the second cohomology of groups,  Schur multiplier, and (projective) representation theory of finite groups. Assuming that participants are familiar with groups, rings, and modules, in the first week, we will develop the basic character theory of representations in characteristic zero along with the concept of cohomology of finite groups. In the second week, we will discuss the projective representations of finite groups and the Schur Multiplier of finite groups in detail.  Representations of GL_2(F_q) will be discussed in adequate detail.