Category Archives: Uncategorized

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 XXXIII

We are back to the convention and We wish to focus on irreducible polynomial representations of G. The weak Peter-Weyl theorem gives: Theorem. Restricting the RHS to only polynomial irreducible V gives us on the LHS, where each polynomial in restricts to a function … Continue reading

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Polynomials and Representations XXXII

We attempt to identify the irreducible rational representations of  From the last article, we may tensor it with a suitable power of det and assume it is polynomial. One key ingredient is the following rather ambiguous statement. Peter-Weyl Principle: any irrep can be embedded inside … Continue reading

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Polynomials and Representations XXXI

K-Representations and G-Representations As mentioned at the end of the previous article, we shall attempt to construct analytic representations of from continuous representations of Let . Consider , where is the group of diagonal matrices in K so as a topological group. From our … Continue reading

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Polynomials and Representations XXX

Representations of GLn and Un Note: all representations of topological groups are assumed to be continuous and finite-dimensional. Here, we will look at representations of the general linear group  We fix the following notations: denotes for some fixed ; is the … Continue reading

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Polynomials and Representations XXIX

Characters Definition. The character of a continuous G-module 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

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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

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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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