We give closed product formulas for the irreducible characters of symmetric groups related to rectangular `almost square' Young diagrams $p \times(p+\delta)$ a fixed value an integer $\delta$ and arbitrary $p$.

We prove that there does not exist a non-square quadratic polynomial with integer coefficients and an axis of symmetry which takes square values for N consecutive integers for N = 7 or N ≥ 9. At the opposite, if N ≤ 6 or N = 8 there are infinitely many.

Abstract. The construction of Freudenthal’s Magic Square, which contains the exceptional simple Lie algebras of types F4, E6, E7 and E8, in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric composition algebras, will be defined, in the split case, in terms of the natural two dimensional module for the simple Lie ...

We provide a purely local computa t ion of the (elliptic) twisted (by "transpose-inverse") character of the representat ion w = I (1) of PGL(3) over a p-adic field induced from the trivial representat ion of the maximal parabolic subgroup. This computa t ion is independent of the theory of the symmetr ic square lifting of [IV] of automorphic and admissible representat ions of SL(2) to PGL(3). I...