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Tag Archives: schur polynomials
Polynomials and Representations XXXV
Schur-Weyl 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 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 XVIII
Littlewood-Richardson Coefficients Recall that the Littlewood-Richardson 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 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 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
Mathematics and Such 