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Tag Archives: young tableaux
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 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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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 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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Polynomials and Representations XV
Tableaux and Words In our context, a word is a sequence of positive integers; concatenation of words is denoted by Given a skew SSYT the corresponding word is obtained by taking the tableau entries from left to right, then bottom to top. For example the … Continue reading
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Tagged knuth equivalence, skew diagrams, sliding algorithm, words, young tableaux
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Polynomials and Representations XIV
In this article, we describe a way of removing the internal squares of a skew SSYT to turn it into an SSYT. Definition. First write the skew diagram as ; we define an inside corner to be a square in such that there is … Continue reading
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Tagged partitions, skew diagrams, sliding algorithm, young tableaux
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