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Tag Archives: schur polynomials
Polynomials and Representations XXXV
SchurWeyl Duality Throughout the article, we denote for convenience. So far we have seen: the Frobenius map gives a correspondence between symmetric polynomials in of degree d and representations of ; there is a correspondence between symmetric polynomials in and polynomial … 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 XXXIII
We are back to the convention and We wish to focus on irreducible polynomial representations of G. The weak PeterWeyl 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 XVIII
LittlewoodRichardson Coefficients Recall that the LittlewoodRichardson coefficient satisfies: By the previous article, for any SSYT of shape , is the number of skew SSYT of shape whose rectification is Since this number is independent of our choice of as long as its shape is … Continue reading
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 nonnegative 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 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 dth 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