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Category Archives: Uncategorized
Polynomials and Representations VIII
Matrix Balls Given a matrix A of non-negative integers, the standard RSK construction masks the symmetry between P and Q, but in fact we have: Symmetry Theorem. If A corresponds to (P, Q), then the transpose of A corresponds to (Q, P). In particular, if A is a … Continue reading
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Tagged combinatorics, matrix balls, partitions, rsk correspondence, young tableaux
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Polynomials and Representations VII
Our next task is as follows: Given partition and vector , count the number of semistandard Young tableaux with shape and type (i.e. occurs times). Proposition. The number of SSYT with shape and type remains invariant when we permute the … Continue reading
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Tagged combinatorics, partitions, rsk correspondence, symmetric polynomials, young tableaux
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Polynomials and Representations VI
For now, we will switch gears and study the combinatorics of the matrices and where run over all partitions of d>0. Eventually, we will show that there is a matrix K such that: where J is the permutation matrix swapping and its transpose. … Continue reading
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Tagged combinatorics, partitions, symmetric polynomials, young tableaux
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Polynomials and Representations V
It was clear from the earlier articles that n (number of variables ) plays a minimal role in the combinatorics of the symmetric polynomials. Hence, removing the parameter n turns out to be quite convenient; the process gives us the formal ring of symmetric functions. … Continue reading
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Tagged combinatorics, partitions, polynomials, symmetric polynomials
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Polynomials and Representations IV
Power Sum Polynomials The power sum polynomial is defined as follows: In this case, we do not define , although it seems natural to set As before, for a partition define: Note that we must have above since we have … Continue reading
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Tagged partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations III
Complete Symmetric Polynomials Corresponding to the elementary symmetric polynomial, we define the complete symmetric polynomials in to be: For example when , we have: Thus, written as a sum of monomial symmetric polynomials, we have Note that while the elementary symmetric polynomials only go … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations II
More About Partitions Recall that a partition is a sequence of weakly decreasing non-negative integers, where appending or dropping ending zeros gives us the same partition. A partition is usually represented graphically as a table of boxes or dots: We will … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations I
We have already seen symmetric polynomials and some of their applications in an earlier article. Let us delve into this a little more deeply. Consider the ring of integer polynomials. The symmetric group acts on it by permuting the variables; specifically, … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Coming up next…
This blog has been dormant for a while, as I’ve been doing quite a bit of self-reading and ruminating over the stuffs I’ve read. I’d really like to post some of my thoughts, but there’s always the risk of misleading … Continue reading
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Mathematics and Such 