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I read on page 4 here that the Kostka coefficients $K_{\lambda,\mu}$ are specializations of the Littlewood-Richardson coefficients $c^\tau_{\sigma,\lambda}$ by specializing $\sigma,\tau$ depending on $\mu$ in a simple manner (certain sums of parts of $\mu$).

Is there a similar specialization/translation for Kostka coefficients obtained from skew shapes, $K_{\lambda,\mu}^\nu$ where $\lambda/\nu$ is a skew shape?

Motivation: I would like to see if polynomiality of the map $n \mapsto K_{n \lambda, nw}^{n \nu}$ implies polynomiality for a similar map with LW-coefficients.

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I prefer to write $K_{\lambda/\nu,\mu}$ for $K^\nu_{\lambda,\mu}$. Using standard symmetric function notation, we have $$ K_{\lambda/\nu,\mu}=\langle s_{\lambda/\nu},h_\mu\rangle = \langle s_\lambda,s_\nu h_\mu\rangle. $$ Let $\rho/\sigma$ be a skew shape which is a disjoint union of shapes $\nu, (\mu_1), (\mu_2), \dots$. Here $(\mu_i)$ is a single row of length $\mu_i$. By "disjoint union," I mean that none of the shapes has a square in the same row or in the same column as a square of another of the shapes. Thus $s_{\rho/\sigma} = s_\nu h_\mu$, so $$ K_{\lambda/\nu,\mu}=\langle s_\lambda,s_{\rho/\sigma}\rangle = \langle s_\lambda s_\sigma, s_\rho\rangle, $$ an ordinary Littlewood-Richardson coefficient.

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  • $\begingroup$ Thanky you, that answers my first question! $\endgroup$
    – Per Alexandersson
    Commented Dec 13, 2012 at 6:51
  • $\begingroup$ I don't really follow the first identity; Macdonald (5.14) says that $K_{\lambda−\mu,\nu}=\langle s_{\lambda/\mu},h_\nu\rangle$ but this is surely different from $K_{\lambda/\mu,\nu}$ which counts the number of skew tableaux of shape $\lambda/\mu$ and weight $\nu$? $\endgroup$
    – Per Alexandersson
    Commented Jun 5, 2013 at 18:59
  • $\begingroup$ $K_{\lambda-\mu,\nu}$ is simply another notation for $K_{\lambda/\mu,\nu}$. In general $\langle f,h_\mu\rangle$ is the coefficient of $m_\mu$ when $f$ is expanded in the basis of monomial symmetric functions. $\endgroup$
    – Richard Stanley
    Commented Jun 5, 2013 at 19:40
  • $\begingroup$ @Richard Stanley Ah, that explains it, but then I must be clearer with stating what I seek; The number of tableaux of shape $\lambda-\mu$ with weight $\nu$, (which I denote $K_{\lambda-\mu,\nu}$) is not the same as the number of skew tableaux of shape $\lambda/\mu$ and weight $\nu,$ which I denote $K_{\lambda/\mu,\nu}.$ (The latter has three free "parameter" partitions, while the first has essentially two.) What I would like is to be able to translate between $K_{\lambda/\mu,\nu}$ and $c_{\tau/\simga,\rho}.$ $\endgroup$
    – Per Alexandersson
    Commented Jun 6, 2013 at 10:42
  • $\begingroup$ Never mind, I think I managed to get the statement that I need, thank you for your effort! $\endgroup$
    – Per Alexandersson
    Commented Jun 6, 2013 at 13:22
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I believe most of what you want is in https://arxiv.org/abs/math/0308101 , especially the polynomiality you're looking for. Note that that was first proven in [H. Derksen, J. Weyman] "On the Littlewood-Richardson polynomials," http://www.math.lsa.umich.edu/~hderksen/preprints/lrpoly.dvi .

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  • $\begingroup$ Yes, I am aware that the function is polynomial in n. The question is if it is easy to see if polynomiality for the Kostka map easily implies polynomiality for LW-coefficients. (The reverse implication should be quite easy, I think). The reason I ask for this, is that I think I have a very short proof of the polynomiality of the map $n \to K_{n\lambda,nw}^{n\mu}$ and it would be interesting to see if this easily implies polynomiality for LW-coefficients. $\endgroup$
    – Per Alexandersson
    Commented Dec 12, 2012 at 20:14
  • $\begingroup$ You mean, again, LR-coefficients, right? The point is that the Steinberg/Klimyk tensor product rule expresses the LR coefficients as an alternating sum (over the Weyl group) of weight multiplicities (here, Kostka numbers). $\endgroup$
    – Allen Knutson
    Commented Dec 13, 2012 at 1:19

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