Two Partial Orders for Standard Young Tableaux

Abstract

In this manuscript we show that two partial orders defined on the set of Littlewood-Richardson fillings of shape β \ γ and content α are equivalent if β \ γ is a horizontal and vertical strip. In fact, we give two proofs for the equivalence of the box order and the dominance order for fillings. Both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second provides a more straightforward construction of box moves. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.MSC 2010: Primary: 05E10, Secondary: 47A15

The boundary of the irreducible components for invariant subspace varieties

“…This completes the proof of Theorem 1.2. Using results given in [13] and in [14], we show part (b) of Theorem 1.3 and complete the proof of Theorem 1.5.…”

“…In this situation, the combinatorial relations ≤ box and ≤ dom are in fact equivalent. In [14] we give two proofs for this statement; below in Section 2.1 we sketch the algorithmic approach in one of them. We deduce the following result.…”

“…However for horizontal and vertical strips, the two partial orders are equivalent [14]: Theorem 2.2. Suppose α, β, γ are partitions such that β \ γ is a horizontal and vertical strip.…”

Box moves on Littlewood–Richardson tableaux and an application to invariant subspace varieties

“…In case the skew diagram β \ γ is a horizontal and vertical strip, the dominance relation is generated by the increasing box moves ( [8]). As a consequence we can describe the relation ≤ boundary : Corollary 5.3 (Corollary 1.2).…”

“…In general, the poset P boundary may be difficult to determine. If, however, the shape β \ γ of the LR-tableaux is a horizontal and vertical strip then it turns out that any two tableaux in dominance partial order can be transformed into each other by using increasing box moves [8], see Section 2.2 for definitions.…”