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Tag Archives: young tableaux
Polynomials and Representations XIII
Skew Diagrams If we multiply two elementary symmetric polynomials and , the result is just , where is the concatenation of and sorted. Same holds for However, we cannot express in terms of easily, which is unfortunate since the Schur functions are 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
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Tagged determinants, partitions, pieri's formula, schur polynomials, young tableaux
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Polynomials and Representations VIII
Matrix Balls Given a matrix A of non-negative integers, the standard RSK construction masks the symmetry between P and Q, but in fact we have: Symmetry Theorem. If A corresponds to (P, Q), then the transpose of A corresponds to (Q, P). In particular, if A is a … Continue reading
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Tagged combinatorics, matrix balls, partitions, rsk correspondence, young tableaux
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Polynomials and Representations VII
Our next task is as follows: Given partition and vector , count the number of semistandard Young tableaux with shape and type (i.e. occurs times). Proposition. The number of SSYT with shape and type remains invariant when we permute the … Continue reading
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Tagged combinatorics, partitions, rsk correspondence, symmetric polynomials, young tableaux
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Polynomials and Representations VI
For now, we will switch gears and study the combinatorics of the matrices and where run over all partitions of d>0. Eventually, we will show that there is a matrix K such that: where J is the permutation matrix swapping and its transpose. … Continue reading
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Tagged combinatorics, partitions, symmetric polynomials, young tableaux
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Mathematics and Such 