Remarks to the dimension of an atypical irreducible representation of the special Lie superalgebra \(\text{sl} (1,n)\) (Q1892769)

The structure of atypical irreducible representations of Lie superalgebras in general, and of \(\text{sl} (m/n)\) in particular, has been a long standing open problem [\textit{V. G. Kac}, Lect. Notes Math. 676, 597-626 (1978; Zbl 0388.17002)]. The author uses some results obtained by \textit{T. D. Palev} [J. Math. Phys. 29, 2589-2598 (1988; Zbl 0782.17003)] describing the structure of atypical representations of \(\text{sl}(1/n)\). As a consequence of this, the problem of giving a dimension formula for \(\text{sl}(1/n)\) is reduced to finding the polynomial solutions of the equation \(g(x_1, \dots, x_n) + g(x_1 - 1, \dots, x_n - 1) = 2^{n + 1} x_1 x_2 \cdots x_n\). The (unique) polynomial solution of this equation is found to be a linear combination of elementary symmetric functions with coefficients \(H_l\), which are numbers closely related to Bernoulli numbers \(B_l\).