Tag Archives: determinants

Polynomials and Representations XXXIX

Posted on July 7, 2016 by limsup

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 →

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Polynomials and Representations XXXVIII

Posted on July 4, 2016 by limsup

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 →

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Polynomials and Representations XII

Posted on May 14, 2016 by limsup

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 →

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Polynomials and Representations XI

Posted on May 13, 2016 by limsup

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 →

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Introduction to Ring Theory (8)

Posted on October 31, 2012 by limsup

Matrix Rings In this post, we’ll be entering the matrix. Let R be a ring. The ring Mn×n(R) is the set of matrices whose entries are elements of R, where the addition and multiplication operations are given by the usual matrix addition … Continue reading →

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