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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

Citations:

Zbl 0699.33012
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References:

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