A Relation for Domino Robinson-Schensted Algorithms

Abstract

We describe a map relating hyperoctahedral Robinson-Schensted algorithms on standard domino tableaux of unequal rank. Iteration of this map relates the algorithms defined by Garfinkle and Stanton-White and when restricted to involutions, this construction answers a question posed by van Leeuwen. The principal technique is derived from operations defined on standard domino tableaux by Garfinkle which must be extended to this more general setting.

Knuth relations for the hyperoctahedral groups

“…Furthermore, by the main result of [15], for these values of r all combinatorial cells are actually independent of r.…”

Sign Under the Domino Robinson-Schensted Maps

“…When r is large relative to n, the relationship between G ∞ and G r is simple and it is a trivial task to translate one sign formula into the other. Using the map of [17], we then extend the result on involutions to all r.…”

“…Our first goal is to verify the theorem for involutions in H n . There are two main tools, the sign formula for colored permutations under G ∞ derived from [15] and a description of the relationship between the maps G r and G r+1 obtained in [17]. When r is large relative to n, the relationship between G ∞ and G r is simple and it is a trivial task to translate one sign formula into the other.…”

“…Next, we extend this formula to all values of r. Let r and r be non-negative integers and suppose that T ∈ SDT r (n). Following [17], we define a map t r,r : SDT r (n) → SDT r (n) by setting t r,r (T ) = T whenever G −1 r (T, T ) = G −1 r (T , T ). When r and r are consecutive integers, this map has a particularly simple description in terms of cycles in a domino tableau.…”

Module structure of cells in unequal-parameter Hecke algebras

“…For T ∈ SDT r (n), we will say that the square s ij is variable if i + j ≡ r mod 2 and fixed otherwise. As discussed in [6] and [22], a choice of fixed squares on a tableau T allows us to define two notions, a partition of its dominos into cycles and the operation of moving through a cycle. The moving through map, when applied to a cycle c in a tableau T yields another standard domino tableau M T (T, c) which differs from T only in the labels of the variable squares of c. If c contains D(l, T ), the domino in T with label l, then M T (T, c) is in some sense the minimally-affected standard domino tableau in which the label of the variable square in D(l, T ) is changed.…”