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One special class of modular forms and group representations. (English) Zbl 0954.11014

The author presents her results on the relationship between eta products and representations of finite groups, most of which had been published in earlier papers. There are exactly 28 eta products \(f(z)= \prod_k \eta^{t_k} (a_kz)\) with positive integers \(a_k\) and \(t_k\) which are cusp forms of integral weight (for some level and character) and which have multiplicative Fourier coefficients. Together with the forms \(\eta(24z)\) and \(\eta^3(8z)\) of half-integral weight we get the list of 30 multiplicative eta products known from D. Dummit, H. Kisilevsky and J. McKay [Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)].
Let \(\Phi\) be a unimodular representation of a finite group \(G\) in a space whose dimension is a multiple of 24. For \(g\in G\), let \(\prod_k(x^{a_k}-1)^{t_k}\) be the characteristic polynomial of \(\Phi(g)\). Then the eta product \(\eta_g(z)= \prod_k\eta^{t_k} (a_kz)\) is associated with \(g\). G. Mason [Contemp. Math. 45, 223-244 (1985; Zbl 0578.10029); Math. Ann. 283, 381-409 (1989; Zbl 0636.10021)] studied representations of the Mathieu group \(M_{24}\) on the Leech lattice and the associated eta products.
The author shows that by means of the adjoint representation all multiplicative eta products of weight \(k\geq 2\) are associated with finite order elements in \(SL(5,\mathbb{C})\). She gives a table of finite subgroups of \(SL(5,\mathbb{C})\) and the corresponding multiplicative eta products. A similar study is given for all groups of order 24 and for dihedral groups.
Finally the author recalls that 16 of the multiplicative eta products are Hecke theta series with groessencharacter on an imaginary quadratic field, and she states identities representing \(\eta^8(z) \eta^8 (2z)\), \(\eta^{12} (2z)\) and \(\eta^{24}(z)\) as series on integral quaternions and on the Cayley algebra, respectively.

MSC:

11F20 Dedekind eta function, Dedekind sums
11F22 Relationship to Lie algebras and finite simple groups
11F11 Holomorphic modular forms of integral weight
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References:

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