Tag Archives: symmetric group

Solving Permutation-Based Puzzles

Introduction In the previous article, we described the Schreier-Sims algorithm. Given a small subset which generates the permutation group G, the algorithm constructs a sequence such that for: we have a small generating set for each Specifically, via the Sims … Continue reading

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Polynomials and Representations XXXVII

Notations and Recollections For a partition , one takes its Young diagram comprising of boxes. A filling is given by a function for some positive integer m. When m=d, we will require the filling to be bijective, i.e. T contains {1,…,d} and each element occurs exactly … Continue reading

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Polynomials and Representations XXXVI

V(λ) as Schur Functor Again, we will denote throughout this article. In the previous article, we saw that the Schur-Weyl duality can be described as a functor: given a -module M, the corresponding -module is set as  Definition. The construction is … Continue reading

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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

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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

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Polynomials and Representations XXVII

From the previous article, we have columns j < j’  in the column tabloid U, and given a set A (resp. B) of boxes in column j (resp. j’), we get:    where is the column tabloid obtained by swapping contents of A with B while preserving the order. … Continue reading

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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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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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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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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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