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There are two versions of Cauchy identity for Schur functions, namely $$ \sum_{\lambda}s_\lambda(\underline x)s_\lambda(\underline y) = \prod_{i,j=1}^n\frac 1{1-x_iy_j}\ ,\qquad {\rm (1)} $$ and $$ \sum_{\lambda}s_\lambda(\underline x)s_{\lambda'}(\underline y) = \prod_{i,j=1}^n (1+x_iy_j)\ .\qquad {\rm (2)} $$

(Usual notations employed here: $\underline x=(x_1,\dots,x_n)$, $\lambda$ runs over all partitions (of length $\leq n$), $\lambda'$ is the conjugate partition.)

I found in the literature (Stanley's 1989 paper, Macdonald's book) the following generalization of (1) to Jack symmetric poylnomials: $$ \sum_{\lambda} \frac{J^{(a)}_ \lambda(\underline x)J^{(a)} _ \lambda(\underline y) }{\langle J^{(a)}_ \lambda,J^{(a)}_ \lambda\rangle_ a}= \prod_{i,j=1}^n\biggl(\frac 1{1-x_iy_j}\biggr)^{1/a}\ . $$

(Here, $J^{(a)}_ \lambda$ are the Jack polynomials (in the $J$-normalization) and $\langle,\rangle_a$ the deformed Hall inner product, namely $\langle p_\lambda,p_\mu\rangle_a=a^{\ell(\lambda)}z_\lambda \delta_{\lambda\mu}$.)

QUESTION: does there exist a similar generalization of (2) for Jack polynomials?

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    $\begingroup$ The second identity you listed is usually called the "dual Cauchy identity." This could help with your searches. $\endgroup$ May 3, 2023 at 13:15
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    $\begingroup$ Thanks! Indeed looking for it, I found it here: kurims.kyoto-u.ac.jp/EMIS/journals/SLC/opapers/s28macdonald.pdf (formula 2.6) $\endgroup$
    – Giulio R
    May 3, 2023 at 14:24
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    $\begingroup$ Just to include the answer directly here: the LHS of (2) changes to $\sum_\lambda P^{(\alpha)}_\lambda(x) \, P^{(1/\alpha)}_{\lambda'}(y)$ for Jack (and $\sum_\lambda P_\lambda(x;q,t) \, P_{\lambda'}(y;t,q)$ for Macdonald, both) in the monic '$P$-normalisation; the RHS remains the same. This is consistent as $\alpha=1$ gives Schur. This identity appears to have been found independently by Gaudin and Macdonald. $\endgroup$ May 3, 2023 at 16:39

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The dual Cauchy identity for Jack polynomials also exists, and is better expressed in terms of the $P$-normalized Jack polynomials: $$ \sum_{\lambda} P^{(a)}_ \lambda(\underline x)P^{(1/a)} _ {\lambda'}(\underline y) = \prod_{i,j=1}^n\bigl(1+x_iy_j\bigr)\ . $$

(Cf., e.g., formula (2.6) in these notes by I.G. Macdonald.)

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    $\begingroup$ I don't think the power on the RHS is right? $\endgroup$ May 3, 2023 at 21:14
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    $\begingroup$ corrected, thanks $\endgroup$
    – Giulio R
    May 3, 2023 at 21:38
  • $\begingroup$ to finish this off you can mark your answer as correct (tick ✓ on the left) $\endgroup$ May 7, 2023 at 21:30

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