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Monthly Archives: June 2016
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 XXXIII
We are back to the convention and We wish to focus on irreducible polynomial representations of G. The weak PeterWeyl 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
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. PeterWeyl Principle: any irrep can be embedded inside … Continue reading
Polynomials and Representations XXXI
KRepresentations and GRepresentations 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
Polynomials and Representations XXX
Representations of GLn and Un Note: all representations of topological groups are assumed to be continuous and finitedimensional. Here, we will look at representations of the general linear group We fix the following notations: denotes for some fixed ; is the … Continue reading
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