# Tag Archives: irreducible representations

## Polynomials and Representations XXXVIII

Determinant Modules We will describe another construction for the Schur module. Introduce variables for . For each sequence we define the following polynomials in : Now given a filling T of shape λ, we define: where is the sequence of entries from the … Continue reading

## Polynomials and Representations XXXVI

V(λ) as Schur Functor Again, we will denote throughout this article. In the previous article, we saw that the Schur-Weyl duality can be described as a functor: given a -module M, the corresponding -module is set as  Definition. The construction is … Continue reading

## Polynomials and Representations XXXV

Schur-Weyl Duality Throughout the article, we denote for convenience. So far we have seen: the Frobenius map gives a correspondence between symmetric polynomials in  of degree d and representations of ; there is a correspondence between symmetric polynomials in and polynomial … Continue reading

## Polynomials and Representations XXXIV

Twisting From the previous article, any irreducible polynomial representation of is of the form for some such that is the Schur polynomial . Now given any analytic representation V of G, we can twist it by taking for an integer k. Then: Twisting the irrep … Continue reading

## Polynomials and Representations XXXI

K-Representations and G-Representations As mentioned at the end of the previous article, we shall attempt to construct analytic representations of from continuous representations of Let . Consider , where is the group of diagonal matrices in K so as a topological group. From our … Continue reading

## Polynomials and Representations XXIX

Characters Definition. The character of a continuous G-module V is defined as: This is a continuous map since it is an example of a matrix coefficient. Clearly for any . The following are quite easy to show: The last equality, … Continue reading

## Quick Guide to Character Theory (II): Main Theory

Reminder: throughout this series, G is a finite group and K is a field. All K-vector spaces are assumed to be finite-dimensional over K. G4. Maschke’s Theorem If is a K[G]-submodule, it turns out V is isomorphic to the direct sum of W and some other submodule W’. … Continue reading