# Tag Archives: young tableaux

## 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 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 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 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