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Results tagged with symmetric-functions
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user 73780
Symmetric functions are symmetric polynomials, in finitely many, or countably infinitely many variables. They arise in the representation theory of symmetric groups and in the polynomial representation theory of general linear groups. Bases of the ring of symmetric functions are indexed by integer partitions. Schur functions, elementary symmetric functions, complete symmetric functions, and power sum symmetric functions are the most commonly used bases.
4
votes
1
answer
118
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Are the minors of this Hadamard product Schur positive?
Let $h_i (x)$ denote the complete symmetric function of degree $i$ in some set of variables $x = (x_1 , x_2 , \dots)$. Then the minors of the Toeplitz matrix $T (x) = \left(h_{i-j} (x) \right)_{i,j}$ …
- 553
2
votes
Accepted
Are the minors of this Hadamard product Schur positive?
The answer to this question is no, with a quick counterexample being given by the determinant of the matrix
$$\begin{pmatrix}
h_{2} (x) h_{2} (y) & h_{3} (x) h_{3} (y) & h_{4} (x) h_{4} (y)\\\\
…
- 553
6
votes
1
answer
79
views
Number of semistandard tableaux of all possible shapes fitting within some rectangle
Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum
$$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$
where $S_\lambda$ denotes the Schur funct …
- 553
2
votes
0
answers
43
views
Formula for sum involving products of (symplectic) Schur functions
This question is a continuation of a question asked yesterday which had a very nice answer.
Consider the summation
$$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\lamb …
- 553