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Monthly Archives: June 2016
Polynomials and Representations XXVI
Let us fix a filling of shape and consider the surjective homomorphism of -modules given by right-multiplying by Specifically, we will describe its kernel, which will have interesting consequences when we examine representations of later. Row and Column Tabloids By the … Continue reading
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Tagged partitions, representation theory, symmetric group, young symmetrizer, young tableaux
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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 i-th row (resp. column) of T to the i-th 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
Power-Sum Polynomials We will describe how the character table of is related to the expansion of the power-sum 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
Mathematics and Such 