Page 27 - Textos de Matemática Vol. 39

JEU DE TAQUIN, INVARIANT FACTORS 17
4. Invariant factors and semi-standard tableaux
Given an n by n non-singular matrix A, with entries in a local principal ideal domain with prime p, by Gaußian elimination one can reduce A to a diagonal matrix ∆μ with diagonal entries pμ1 , . . . , pμn , for unique nonnegative integers μ1 ≥ . . . ≥ μn , called the Smith normal form of A. The sequence pμ1 , . . . , pμn defines the invariant factors of A, and (μ1, . . . , μn) is the invariant partition of A. It is known that μ, β, γ are invariant partitions of nonsingular matrices A, B, and C such that AB = C if and only if there exists a Littlewood- Richardson tableau T of type (μ,β,γ), that is, a SSYT of shape γ/μ whose word is in the Knuth class of the key tableau of weight β (Yamanouchi tableau of weight β). Apart from other approaches [12, 25, 15, 16], this result can be derived in a purely matrix context when one introduces the following
Definition 4.1. [3] Let T = (μ0, μ1, ..., μt) and F = (0, δ1, ..., δt) be SSYT’s both of weight (m1, ...,mt). We say that a sequence of n by n nonsingular matrices A0,B1,...,Bt is a matrix realization of the pair (T,F) of SSYT’s if:
I. For each r ∈ {1, ..., t}, the matrix Br has invariant partition (1mr , 0n−mr ). II. For each r ∈ {0, 1, ..., t}, the matrix Ar := A0B1...Br has invariant parti-
tion the conjugate of μr.
III. For each r ∈ {1,...,t}, the matrix B1...Br has invariant partition the
conjugate of δr.
Given J ⊆ [n], we write DJ for the diagonal matrix having the ith diag- onal entry equal to p whenever i ∈ J and 1 otherwise. For the Bender-Knuth involution we have the following interpretation.
Theorem 4.2. [3, 5] Let T be a SSYT with inner shape μ, indexing sets J1 , J2 , andwordω.Letμ=(al1,al2,...,alk),a1 >···>ak,li >0,1≤i≤k,and
12k
Xq = {q−1 lj + 1, . . . , q lj } with l0 = 0 and 1 ≤ q ≤ k. If T ′ is the SSYT
j=0 j=0
realized by ∆μ, DJ2 , DJ1 with indexing sets J1′ , J2′ and word ω′, then
(a)(J2′ ∩Xq)(J1′ ∩Xq)=Θ1(J2∩Xq)(J1∩Xq),1≤q≤k.
(b) ωq′ = t1ωq, where ωq and ωq′ are the subwords, respectively, of ω and ω′ restricted to the positions in Xq, 1 ≤ q ≤ k.
T′ is the result of the application of the evacuation to T [5, 17]. In fact we have
and
∆μDJ1 DJ2 = ∆μDJ2 DJ1 =
k q=1
k q=1
∆aq DJ1∩Xq DJ2∩Xq , ∆aq DJ2∩Xq DJ1∩Xq .