# Monthly Archives: May 2016

## 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

## Polynomials and Representations XVIII

Littlewood-Richardson Coefficients Recall that the Littlewood-Richardson coefficient satisfies: By the previous article, for any SSYT of shape ,  is the number of skew SSYT of shape whose rectification is Since this number is independent of our choice of as long as its shape is … Continue reading

## Polynomials and Representations XVII

Two Important Results In this article and the next, we will find a combinatorial way of computing the Littlewood-Richardson coefficient. The key result we have so far is that given any word w there is a unique SSYT T (called the rectification of … Continue reading

## Polynomials and Representations XVI

Here is the main problem we are trying to solve today. Word Problem Given a word let us consider disjoint subwords of which are weakly increasing. For example if , then we can pick two or three disjoint subwords as follows: For … Continue reading

## Polynomials and Representations XV

Tableaux and Words In our context, a word is a sequence of positive integers; concatenation of words is denoted by Given a skew SSYT  the corresponding word is obtained by taking the tableau entries from left to right, then bottom to top. For example the … Continue reading

## Polynomials and Representations XIV

In this article, we describe a way of removing the internal squares of a skew SSYT to turn it into an SSYT. Definition. First write the skew diagram as ; we define an inside corner to be a square in such that there is … Continue reading