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

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

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So suppose > 0. Since < (k) , (k−1) (k−1) < (k) and m (k) −1 (ν (k−1) ) = 0, one finds ≤ and (k−1) ≤ . Also (k) there is a string of length in ν , which is both singular and has label zero since (k) (k) (k) P (ν) = 0. But < (k) ≤ , which contradicts the definition of and (k) . 11) fails for (k) − 1. 3) and hence the proof of the generic case for ν (k) . Doubly singular case. Since there is a string of length in ν (k) that is both (k) singular and has label zero, it must have vacancy number zero, that is, P (ν) = 0.

Schilling and M. Shimozono and m (k−1) −1 (ν (k) ) = 0, that is, m that m −1 (ν (k+1) ) = 0, as desired. Otherwise P (k) −1 (ν) −1 (ν (k) ) = 0. 10) for n = − 1, it follows = 1. Here ≥ 2. 6), (k−1) (k) (k) Sel. , New ser. ≤ − 1 and (k−1) ≤ − 1. (k) and , there cannot be strings in ν of length − 1 By the minimality of that are singular or have label zero, so m −1 (ν (k) ) = 0. 10) at n = −1 (k) and using the fact that P (ν) = 0 (since the doubly singular case holds for ν (k) ) one obtains P (k) −2 (ν) +m −1 (ν (k−1) )+m −1 (ν (k+1) ) ≤ 2.

L CLR(λ; R) into column-strict tableaux. 9. Statistics for LR tableaux and rigged configurations Recall that both the set of LR tableaux and the set of rigged configurations are endowed with statistics, given by the charge cR (T ) for T ∈ CLR(λ; R) and cc(ν, J) for (ν, J) ∈ RC(λt ; Rt ). 1)]. The objective of this section is to prove that φ preserves the statistics, which settles [13, Conjecture 9]. 1. Let T ∈ CLR(λ; R). Then cR (T ) = cc(φR (T )). Proof. 5. By [24, Prop. 7], for all S ∈ CLR(λ; R) we have crows(R) (iR (S)) = cR (S).

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