## Intersection matrix of a generalized Airy function in terms of skew-Schur polynomials.(English)Zbl 0971.33006

Let $$f(a,t) = \sum_{k=0}^N(-1)^k e_k(a)\theta_{N-k+1}(t)$$ be the polynomial of $$t=(t_1, \dots, t_n)$$ of degree $$N+1$$ given in terms of elementary symmetric polynomials $$e_k(a)$$ of $$a=(a_1, \dots, a_N)$$ and $$\log(1+ t_1 X + \cdots + t_n X^n) = \sum_{k=1}^\infty \theta_k(t)X^k.$$ Let $A(a) = \int_c e^{f(a,t)} \omega$ be a generalization of the Airy integral introduced by I. M. Gel’fand, V. S. Retakh and V. V. Serganova [Sov. Math., Dokl. 37, No. 1, 8–12 (1988; Zbl 0699.33012)], where $$\omega$$ is an $$n$$-form representing an $$n$$-cocycle of the polynomial de Rham cohomology group $$H^n( \Omega, d_f)$$ with the differential $$d_f= d + df\wedge$$ and $$c$$ is an appropriate $$n$$-cycle. We see that $$\dim H^n(\Omega, d_{\pm f}) = \binom{N}{n}$$ and that a basis $$\{\phi_\lambda \}$$ can be given by $\phi_\lambda = s_\lambda(z) dt_1 \wedge \cdots \wedge dt_n$ indexed by the Young subdiagram $$\lambda$$ of the rectangular Young diagram $$R(n, N-n)$$ of $$n$$-rows and $$(N-n)$$-columns, where $$s_\lambda(z)$$ is the Schur polynomial of $$z=(z_1, \dots ,z_n)$$ which is regarded as a polynomial of $$t_1=e_1(z), \dots ,t_n=e_n(z)$$. The authors evaluate explicitly the cohomological intersection pairing $H^n(\Omega, d_f) \times H^n(\Omega, d_{-f}) \longrightarrow {\mathbb C}$ for the bases $$\{ \phi_\lambda ^+\}$$ and $$\{ \phi_\lambda ^-\}$$ given by the forms $$\phi_\lambda$$. In fact they gave the formula $\langle \phi_\lambda^+, \phi_\mu^- \rangle = (-1)^{n(n-1)/2} n ! s_{\lambda/\breve{\mu }},$ where $$s_{\lambda/\breve{\mu}}$$ denotes the skew-Schur polynomial for $$\lambda$$ and the complementary diagram $$\breve{\mu}$$ of $$\mu$$ in $$R(n, N-n).$$

### MSC:

 33C70 Other hypergeometric functions and integrals in several variables 14F40 de Rham cohomology and algebraic geometry 32S20 Global theory of complex singularities; cohomological properties

Zbl 0699.33012
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### References:

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