An unpublished theorem of Manfred Schocker and the Patras-Reutenauer algebra

Abstract

Patras, Reutenauer (J. Algebr. Comb. 16:301–314, [2002]) describe a subalgebra \(\mathfrak{A}\) of the Malvenuto-Reutenauer algebra ℘. Their paper contains several characteristic properties of this subalgebra. In an unpublished manuscript Manfred Schocker states without proof a theorem, providing two further characterizations of the Patras-Reutenauer algebra. In this paper we establish a slightly generalized version of Schocker’s theorem, and give some applications. Finally we describe a derivation of the convolution algebra \(\mathfrak{A}\) , which is a homomorphism for the inner product.