A bijection between Littlewood-Richardson tableaux and by Kirillov A.N., Schilling A., Shimozono M.

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By Kirillov A.N., Schilling A., Shimozono M.

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Hence assume length −2 in ν , which contradicts the minimality of (k) and (k) (k) that p < − 2. 11, Pn (ν) = 1 for p + 1 ≤ n ≤ − 1 and Pp (ν) ≤ 1. 10) mn (ν (k−1) ) = mn (ν (k+1) ) = 0 for p + 2 ≤ n ≤ − 2 and Pp(k) (ν) + mp+1 (ν (k−1) ) + mp+1 (ν (k+1) ) ≤ 1. 22) (k) Suppose Pp (ν)=1. This case can only occur if p≥1. 22) mp+1 (ν (k−1) ) = (k−1) 0, so that (k−1) ≤ p and ≤ p. But mp (ν (k) ) ≥ 1, so there is either a string (k) of length p < that is singular or of label 0, contradicting the minimality in ν (k) (k) of and .

The case (E1) Suppose R R+ as in (E1). Define the embedding + : RC(λt ; Rt ) → RC(λt ; R+t ) by the commutativity of the diagram ✲ RC(λt ; R+t ) RC(λt ; Rt ) inclusion θR + θR ❄ RC(λt ; Rt ) ❄ ✲ RC(λt ; R+t ). 3) Vol. 8 (2002) Bijection between LR tableaux and rigged configurations 109 Note that + preserves colabels. 3 in the case (E1). 4. The diagram commutes: ı+ ✲ CLR(λ; R) CLR(λ; R+ ) φR+ φR ❄ RC(λt ; Rt ) ❄ ✲ RC(λt ; R+t ). +  Proof. 1 with R< (resp. ı< , < ) replaced by R+ (resp. ı+ , + ).

Recall that ξ (k) (R) is the partition whose parts are the heights of the rectangles in R of width R k. Say that R R if ξ (k) (R) ξ (k) (R ) for all k ≥ 1. Clearly R R and R if and only if R is a reordering of R. Thus the relation R R is a preorder. 3]) (E1) R R2+ (E2) R the R+ where Ri = Ri+ for i > 2, R1 = (ca ), R2 = (cb ), R1+ = (ca−1 ), = (cb+1 ) for a − 1 ≥ b + 1 and c a positive integer. sp R where sp R denotes the sequence obtained from R by exchanging rectangles Rp and Rp+1 . 1) ❄ ✲ CLR(λt ; R+t ).

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