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Category Archives: Uncategorized
Polynomials and Representations XXV
Properties of the Young Symmetrizer Recall that for a filling , we have the subgroup of elements which take an element of the ith row (resp. column) of T to the ith row (resp. column) of T. Then: where is the Young symmetrizer. … Continue reading
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Tagged partitions, representation theory, symmetric group, young symmetrizer, young tableaux
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Polynomials and Representations XXIV
Specht Modules Till now, our description of the irreps of are rather abstract. It would be helpful to have a more concrete construction of these representations – one way is via Specht modules. First write Thus if , the only common irrep between … Continue reading
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Tagged group actions, representation theory, symmetric group, young symmetrizer, young tableaux
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Polynomials and Representations XXIII
PowerSum Polynomials We will describe how the character table of is related to the expansion of the powersum symmetric polynomials in terms of monomials. Recall: where exactly since is not defined. Now each irrep of is of the form for some … Continue reading
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Tagged character theory, combinatorics, partitions, symmetric group, symmetric polynomials
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Polynomials and Representations XXII
Product of Representations Recall that the Frobenius map gives an isomorphism of abelian groups: Let us compute what the product corresponds to on the RHS. For that, we take and where and Multiplication gives where is the partition obtained by sorting Next, we … Continue reading
Polynomials and Representations XXI
We have established that all irreps of are defined over and hence any field of characteristic 0. For convenience we will fix . Twists For any group G and representation over if is a group homomorphism, we can twist as follows: Sometimes, we also … Continue reading
Polynomials and Representations XX
From now onwards, we will assume the base field K has characteristic 0. Example: d=3 Following the previous article, we examine the case of . We get 3 partitions: , and Let us compute for all From the previous article, we have: Since , is … Continue reading
Polynomials and Representations XIX
Representations of the Symmetric Group Let [d] be the set {1,…,d}, and Sd be the group of bijections From here on, we shall look at the representations of Note that this requires a good understanding of representation theory (character theory) of finite groups. To start, let … Continue reading
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Tagged character theory, group actions, representation theory, symmetric group
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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 XVII
Two Important Results In this article and the next, we will find a combinatorial way of computing the LittlewoodRichardson coefficient. The key result we have so far is that given any word w there is a unique SSYT T (called the rectification of … Continue reading
Polynomials and Representations XVI
Here is the main problem we are trying to solve today. Word Problem Given a word let us consider disjoint subwords of which are weakly increasing. For example if , then we can pick two or three disjoint subwords as follows: For … Continue reading
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Tagged skew diagrams, sliding algorithm, word problem, words, young tableaux
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