# Tag Archives: symmetric polynomials

## 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 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 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 XXIII

Power-Sum Polynomials We will describe how the character table of is related to the expansion of the power-sum symmetric polynomials in terms of monomials. Recall: where exactly since is not defined. Now each irrep of is of the form  for some … 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 X

Cauchy’s Identity In this article, our primary focus is the ring of symmetric polynomials in Theorem (Cauchy’s Identity). Consider polynomials over all partitions [Recall that  if ] We have an equality of formal power series: Note. For convenience, we will use  … Continue reading

## Polynomials and Representations IX

Hall Inner Product Let us resume our discussion of symmetric polynomials. First we define an inner product on d-th component of the formal ring. Recall that the sets are both -bases of . Definition. The Hall inner product is defined by setting and to be … Continue reading