Swiching Rule on the Shifted Rim Hook Tableaux

When the Schur function sλ corresponding to a partition λ is defined as the generating function of the column strict tableaux of shape λ it is not at all obvious that sλ is symmetric. In [BK] Bender and Knuth showed that sλ is symmetric by describing a switching rule for column strict tableaux, which is essentially equivalent to the jeu de taquin of Schützenberger (see [Sü]). Bender and Knuth’s switching rule shows that the number of column strict tableaux of a given shape is independent of the order of the contents. Stanton and White[SW2] gave a rim hook analog of this switching procedure. In this paper we describe a switching algorithm for shifted rim hook tableaux, which shows that the sum of the weights of the shifted rim hook tableaux of a given shape and content does not depend on the order of the content if content parts are all odd. Using the recurrence formula for the irreducible spin characters of S̃n , this will show that φ ρ = φ ρ′ , where ρ has all odd parts and ρ ′ is any reordering of ρ. In section 1, we outline the definitions and notation used in this paper. In section 2, we review the basic properties of a group S̃n and draw some relations between the irreducible spin characters of S̃n and symmetric functions. In section 3, a swiching rule on the shifted rim hook tableaux is given.