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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 polynomials, these constants are called the Littlewood-Richardson coefficients. What are they called for monomial symmetric polynomials, and how do I calculate them?

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    I once h$a$d $a$ similar [question](http://math.stackexchange.com/q/83214/19341) with a very nice answer.2012-02-28

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I found this reference, where the authors deal with the products you asked for.


EDIT The reference is

A MAPLE program for calculations with Schur functions by M.J. Carvalho, S. D’Agostino Computer Physics Communications 141 (2001) 282–295


From the paper (p.5 chap. 3.1 Multiplication and division of $m$-functions):

Let’s define the result of the addition and subtraction of two partitions $(\mu_1,\mu_2, . . .)$ and $(\nu_1, \nu_2, . . .)$ as being the partition whose parts are $(\mu_1 ± \nu_1,\mu_2 ± \nu_2, . . .)$. For these operations to be meaningful, it is necessary that both partitions have an equal number of parts; if they do not, then one increases the number of parts of the shortest one by adding enough zeros at the end. ... The multiplication (and division) of two m-functions are then defined as $ m_{\alpha} m_{\beta} = \Sigma I_{\gamma}m_{\gamma} $ and $ m_{\alpha}/ m_{\beta} = \Sigma I_{\gamma'}m_{\gamma'} $ where the partitions $\gamma$,$\gamma'$ result from adding to or subtracting, respectively, from $\alpha$ all distinct partitions obtained by permuting in all possible ways the parts of $\beta$. Clearly, all $m$-functions involved are functions of the same $r$ indeterminates, i.e. have the same number of total parts. The coefficient $I_\nu$, with $\nu = \gamma$ is given by $ I_\nu=n_\nu \frac{\dim (m_\alpha)}{\dim (m_\nu)} $ where $n_\nu$ is the number of times the same partition $\nu$ appears in the process of adding or subtracting partitions referred to above.

As far as I read, they don't give a special name to these coefficients.

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    Reading the answer again, I have strong doubts about the division case. What do they mean by that? Note that the monomial function $m_{2,1}=e_2e_1-3e_3$ corresponds to an irreducible polynomial in the polynomial ring $\Bbb C[e_1,e_2,e_3,\ldots]$ and I think this is typical for the $m_\alpha$. But then they cannot be divisible by any other $m_\beta$, and the quotient $m_\alpha/m_\beta$ can only make sense as a formal rational fraction, not a polynomial as the RHS of the equation indicates. So either they are using a curious definition of division, or this is just plain false.2013-05-22