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 \(R(n, N-n).\)


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
Full Text: DOI Euclid


[1] Airy, G. B.: On the intensity of light in the neighborhood of a caustic. Trans. Cambridge Phil. Soc., 6 , 379-402 (1838).
[2] Aomoto, K., and Kita, M.: Hypergeometric functions. Springer, Tokyo (1994). · Zbl 1229.33001
[3] Fulton, W., and Pragacz, P.: Schubert varieties and degeneracy loci. Lecture Notes in Math., 1689 , Springer, Berlin (1998). · Zbl 0913.14016
[4] Gel’fand, I. M., Retakh, V. S., and Serganova, V. V.: Generalized Airy functions, Schubert cells and Jordan groups. Dokl. Akad. Nauk SSSR, 298 , 17-21 (1988); Soviet Math. Dokl., 37 , 8-12 (1988) (English transl.). · Zbl 0699.33012
[5] Iwasaki, K.: Isolated singularity, Witten’s Laplacian, and duality for twisted de Rham cohomology (2000) (preprint). · Zbl 1032.58027 · doi:10.1081/PDE-120019373
[6] Iwasaki, K., and Kita, M.: Exterior power structure on the twisted de Rham cohomology of the complements of real Veronese arrangements. J. Math. Pures Appl., 75 , 69-84 (1995). · Zbl 0853.58004
[7] Kimura, H.: On rational de Rham cohomology associated with generalized Airy functions. Ann. Scuola Norm. Sup. Pisa, Cl. Sci., (4) 24 , 351-366 (1997). · Zbl 0887.33007
[8] Kimura, H.: On the homology group associated with the general Airy integral. Kumamoto J. Math., 10 , 11-29 (1997). · Zbl 0898.33001
[9] Kita, M., and Yoshida, M.: Intersection theory for twisted cycles I, II. Math. Nachr., 166 , 287-304 (1994); 168 , 171-190 (1994). · Zbl 0847.32043 · doi:10.1002/mana.19941660122
[10] Macdonald, I. G.: Symmetric functions and Hall polynomials. 2nd ed., Clarendon Press, Oxford (1995). · Zbl 0824.05059
[11] Matsumoto, K.: Intersection numbers for \(1\)-forms associated with confluent hypergeometric functions. Funkcial. Ekvac., 41 , 291-308 (1998). · Zbl 1140.33303
[12] Matsumoto, K., and Yoshida, M.: Recent progress of intersection theory for twisted (co)homology groups. (eds. Falk, M. J., and Terao, H.) Arrangements: Tokyo 1998, Adv. Stud. Pure Math., Kinokuniya, Tokyo 1998, pp. 217-237 (2000). · Zbl 0971.55010
[13] Pham, F.: Vanishing homologies and the \(n\)-variable saddle point method. Proc. Symp. Pure Math., 40 , 319-333 (1983). · Zbl 0519.49026
[14] Pham, F.: La descente des cols par les onglets de Lefschetz avec vues sur Gauss-Manin. Astérisque, 130 , 11-47 (1985). · Zbl 0597.32012
[15] Yoshida, M.: Intersection theory for twisted cycles III - Determinant formulae, Math. Nachr., 214 , 173-185 (2000). · Zbl 0958.32024 · doi:10.1002/1522-2616(200006)214:1<173::AID-MANA173>3.0.CO;2-0
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