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Tag Archives: partitions
Polynomials and Representations XXXVII
Notations and Recollections For a partition , one takes its Young diagram comprising of boxes. A filling is given by a function for some positive integer m. When m=d, we will require the filling to be bijective, i.e. T contains {1,…,d} and each element occurs exactly … 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 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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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 i-th row (resp. column) of T to the i-th 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 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 XIV
In this article, we describe a way of removing the internal squares of a skew SSYT to turn it into an SSYT. Definition. First write the skew diagram as ; we define an inside corner to be a square in such that there is … Continue reading
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Tagged partitions, skew diagrams, sliding algorithm, young tableaux
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Polynomials and Representations XI
Here, we will give a different interpretation of the Schur polynomial, however this definition only makes sense in the ring For a given vector of non-negative integers, define the following determinant, a polynomial in : For the case where , we … Continue reading
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Tagged determinants, partitions, pieri's formula, schur polynomials, young tableaux
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Polynomials and Representations X
Cauchy’s Identity In this article, our primary focus is the ring of symmetric polynomials in Theorem (Cauchy’s Identity). Consider polynomials over all partitions [Recall that if ] We have an equality of formal power series: Note. For convenience, we will use … Continue reading
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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Mathematics and Such 