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7
votes
2answers
513 views
Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?
Given the elementary symmetric polynomials $e_k(X_1,X_2,...,X_N)$ generated via
$$
\prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N.
$$
How can one get the monomial symmetric functions ...
3
votes
2answers
504 views
Scalar Product for Vector Space of Monomial Symmetric Functions
Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$:
$$
P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} .
$$
Since ...
-3
votes
3answers
388 views
How do I find out the symmetry of a function?
For example, how do I know that with:
$$f(x_1,x_2,x_3,x_4)=\frac{x_1 x_2+x_3 x_4-x_2 x_3-x_1 x_4}{x_1 x_2+x_3 x_4-x_1 x_3-x_2 x_4}$$
$f$ has the property:
...
5
votes
2answers
111 views
Symmetry of a Plücker function
Let $d \in \mathbb{N}$ and let $I$ be a set. Let $\omega : I^d \times I^d \to \mathbb{R}$ be a function, denoted by $(a_1,\dotsc,a_d,b_1,\dotsc,b_d) \mapsto a_1 \cdots a_d | b_1 \cdots b_d$, with the ...
5
votes
0answers
116 views
Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.
(this is an improved version of What about other symmetric functions of the eigenvalues? )
Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
96 views
Schur skew functions
Let $\lambda,\mu,\nu$ be some partitions. Let's denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If
$s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$
...
9
votes
1answer
142 views
What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials?
The product of monomial symmetric polynomials can be expressed as
$m_{\lambda} m_{\mu} = \Sigma c_{\lambda\mu}^{\nu}m_{\nu}$
for some constants $c_{\lambda\mu}^{\nu}$.
In the case of Schur ...
1
vote
0answers
33 views
decomposition of products of monomial symmeric polynomials into sums of them
I'm trying to make sense of the answer given in: this question
I am stuck at the phrase 'where the partitions γ result from adding, respectively, from α all distinct partitions obtained by permuting ...
