# 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 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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+1 very interesting question – draks ... Feb 27 '12 at 22:11
I once had a similar question with a very nice answer. – draks ... Feb 28 '12 at 14:21

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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Not mentioning the details of the reference in the answer, but only (supposedly) on the other end of a link is not a great idea. The link is dead now, and I have no idea where it used to be pointing. – Marc van Leeuwen May 19 '13 at 13:22
And is "subtracting" missing after "respectively"? Adding $\alpha$ from something else is bad syntax... – Marc van Leeuwen May 19 '13 at 13:27
@Marc thanks for reporting that. I try to fix the link (if possible) and provide more information about the paper in the body of the post...do you have any special interest in the topic? – draks ... May 19 '13 at 13:34
@marc sorry you were right. I stripped the reference a bit to fit to the question. Stand by... – draks ... May 19 '13 at 21:39
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. – Marc van Leeuwen May 22 '13 at 6:23