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Tag Archives: combinatorics
Free Groups and Tiling
Introduction Consider the following simple problem. Prove that the shape on the left cannot be completely tiled by 20 polygons of the types shown on the right. The solution is rather simple: colour the shape in the following manner. This … Continue reading
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Tagged combinatorics, free groups, group theory, groups, polyominoes, tiling, words
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Polynomials and Representations XXIII
Power-Sum Polynomials We will describe how the character table of is related to the expansion of the power-sum symmetric polynomials in terms of monomials. Recall: where exactly since is not defined. Now each irrep of is of the form for some … Continue reading
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Tagged character theory, combinatorics, partitions, symmetric group, symmetric polynomials
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Polynomials and Representations IX
Hall Inner Product Let us resume our discussion of symmetric polynomials. First we define an inner product on d-th component of the formal ring. Recall that the sets are both -bases of . Definition. The Hall inner product is defined by setting and to be … Continue reading
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 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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