# Tag Archives: character theory

## Polynomials and Representations XXXIX

Some Invariant Theory We continue the previous discussion. Recall that for we have a -equivariant map which induces an isomorphism between the unique copies of in both spaces. The kernel Q of this map is spanned by for various fillings T with shape and entries … Continue reading

## 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 XXXIII

We are back to the convention and We wish to focus on irreducible polynomial representations of G. The weak Peter-Weyl theorem gives: Theorem. Restricting the RHS to only polynomial irreducible V gives us on the LHS, where each polynomial in restricts to a function … Continue reading

## Polynomials and Representations XXXII

We attempt to identify the irreducible rational representations of  From the last article, we may tensor it with a suitable power of det and assume it is polynomial. One key ingredient is the following rather ambiguous statement. Peter-Weyl Principle: any irrep can be embedded inside … 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 XXX

Representations of GLn and Un Note: all representations of topological groups are assumed to be continuous and finite-dimensional. Here, we will look at representations of the general linear group  We fix the following notations: denotes for some fixed ; is the … 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