# Tag Archives: determinants

## 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 XII

Lindström–Gessel–Viennot Lemma Let us switch gears and describe a beautiful combinatorial result. Suppose is a graph which is directed, has no cycles, and there are only finitely many paths from a vertex to another. Given sets of n vertices: the lemma … Continue reading

## Polynomials and Representations XI

Here, we will give a different interpretation of the Schur polynomial, however this definition only makes sense in the ring For a given vector of non-negative integers, define the following determinant, a polynomial in : For the case where , we … Continue reading