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

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Define ı< : CLR(λ; R) → CLR(λ; R< ) as ı< := trLR ◦ı∨ ◦ trLR . 100 A. N. Kirillov, A. Schilling and M. Shimozono Sel. , New ser. t Note that R> = (Rt )∧ . Similarly define ı> : CLR(λ; R) → CLR(λ; R> ) as ı> := trLR ◦ı∧ ◦ trLR . 2) ❄ ev ✲ CLR(λ; R> ). < ı t Let > : RC(λt ; Rt ) → RC(λt ; R> ) be the inclusion map. Then define < by the following commutative diagram: RC(λt ; Rt ) > ✲ t RC(λt ; R> ) ev θR > ev θR ❄ RC(λt ; Revt ) ❄ ✲ RC(λt ; R> evt ). 3) For (ν, J) ∈ RC(λt ; Rt ) set (ν > , J > ) = > (ν, J).

Suppose neither holds. Then = . 1) (k−1) = . Now m (ν (k−1) ) = 1 and = = (k−1) (k) ≥ (k−1) > −1 so m (ν (k−1) ) = 0, = . contradicting (k) Suppose m (ν (k−1) ) = 0. 1), it follows that (k−1) ≤ − 1 and ≤ − 1. Suppose (k−1) (k−1) = . Now m (ν m (ν (k−1) ) = 0. 17). 16) we have (k) (k−1) (k) ≤ − 1. 12) we have > − 1. Since (k−1) ) = 0 and (k−1) (k−1) ≤ ≤ (k) = − 1 so = . (k) Pp−1 (ν) = Pp−1 (ν) − χ( > (k) (k−1) and (k+1) ≥ ≤ p − 1) + 1. 9) 126 A. N. Kirillov, A. Schilling and M. 1), (k) = ≥ (k−1) (k−1) Sel.

N. Kirillov, A. Schilling and M. Shimozono Sel. , New ser. 3. 6) to express φR inductively in terms of φR∧ . 5) to express φR in terms of φR> . It will be shown in Section 7 that this recurrence, which also defines φR , is in a sense transpose to the usual definition of the bijection φR t . 7. 7. An analogous RC-transpose bijection exists for the set of rigged configurations denoted by trRC : RC(λt ; Rt ) → RC(λ; R), which was described in [13, Section 9]. 1. 1) mij = αj (i) for i, j ≥ 1, where αj is the size of the j-th column of the partition ν (i) , recalling that ν (0) is defined to be the empty partition.

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

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