# Tag Archives: representation theory

## 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

## Polynomials and Representations XXVIII

Starting from this article, we will look at representations of . Now, itself is extremely complicated so we will only focus on representations of particular types. Generally, for any topological group G, we want: to be a continuous homomorphism of groups. Continuous … Continue reading

## Polynomials and Representations XXVII

From the previous article, we have columns j < j’  in the column tabloid U, and given a set A (resp. B) of boxes in column j (resp. j’), we get:    where is the column tabloid obtained by swapping contents of A with B while preserving the order. … Continue reading

## Polynomials and Representations XXVI

Let us fix a filling of shape and consider the surjective homomorphism of -modules given by right-multiplying by Specifically, we will describe its kernel, which will have interesting consequences when we examine representations of later. Row and Column Tabloids By the … Continue reading

## Polynomials and Representations XXV

Properties of the Young Symmetrizer Recall that for a filling , we have the subgroup of elements which take an element of the i-th row (resp. column) of T to the i-th row (resp. column) of T. Then: where  is the Young symmetrizer. … Continue reading

## Polynomials and Representations XXIV

Specht Modules Till now, our description of the irreps of are rather abstract. It would be helpful to have a more concrete construction of these representations – one way is via Specht modules. First write Thus if , the only common irrep between … Continue reading

## Polynomials and Representations XXII

Product of Representations Recall that the Frobenius map gives an isomorphism of abelian groups: Let us compute what the product corresponds to on the RHS. For that, we take and where and Multiplication gives where is the partition obtained by sorting Next, we … Continue reading

## Polynomials and Representations XXI

We have established that all irreps of are defined over and hence any field of characteristic 0. For convenience we will fix . Twists For any group G and representation over  if is a group homomorphism, we can twist as follows: Sometimes, we also … Continue reading

## Polynomials and Representations XX

From now onwards, we will assume the base field K has characteristic 0. Example: d=3 Following the previous article, we examine the case of . We get 3 partitions: , and Let us compute for all From the previous article, we have: Since , is … Continue reading

## Polynomials and Representations XIX

Representations of the Symmetric Group Let [d] be the set {1,…,d}, and Sd be the group of bijections  From here on, we shall look at the representations of Note that this requires a good understanding of representation theory (character theory) of finite groups. To start, let … Continue reading