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

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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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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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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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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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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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