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Tag Archives: representation theory
Polynomials and Representations XXIX
Characters Definition. The character of a continuous Gmodule V is defined as: This is a continuous map since it is an example of a matrix coefficient. Clearly for any . The following are quite easy to show: The last equality, … Continue reading
Polynomials and Representations XXVIII
Starting from this article, we will look at representations of . Now, itself is extremely complicated so we will only focus on representations of particular types. Generally, for any topological group G, we want: to be a continuous homomorphism of groups. Continuous … 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 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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