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Tag Archives: symmetric group
Solving PermutationBased Puzzles
Introduction In the previous article, we described the SchreierSims 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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Tagged group actions, group theory, permutations, rubik's cube, schreiersims, symmetric group
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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
Polynomials and Representations XXXVI
V(λ) as Schur Functor Again, we will denote throughout this article. In the previous article, we saw that the SchurWeyl duality can be described as a functor: given a module M, the corresponding module is set as Definition. The construction is … Continue reading
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 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
Polynomials and Representations XXVI
Let us fix a filling of shape and consider the surjective homomorphism of modules given by rightmultiplying 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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