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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Since = one part The assumption of size is shortened in passing from ν (k) to ν (k) , so that m (ν (k) ) ≥ 2. Now (k) P (ν) = 0, so there is at least one singular string in ν (k) that is not selected by δ acting on (ν, J). 1 this string remains singular of length in passing to ν (k) . This shows that there is a singular string of length in ν (k) . Thus to prove (k) = If it suffices to show that (k−1) ≤ (k−1) , then so Case 3 holds for ν (k−1) (k−1) (k−1) ≤ . (k−1) ≤ , and by induction ev this also shows that if By applying θR Case 3: (k) = (k) (k) ≤ (k) = .

1 with R< (resp. ı< , < ) replaced by R+ (resp. ı+ , + ). 1. 3. 3 is an immediate consequence of the following result. 5. 4) ❄ RC(λt ; (sp R)t ). Proof. Recall the intervals of integers Bj (1 ≤ j ≤ L) in the definition of the set CLR(λ; R). It follows from the definition of σp that for all S ∈ CLR(λ; R), (σp S)|Bj = S|Bj for j > p + 1. Using this fact and the definition of φR , one may reduce to the case L = p + 1. So it may be assumed that L = p + 1. Obviously (sp R)ev = s1 (Rev ). Consider the diagram CLR(λ; R) yyy yyyev yyy yy9 CLR(λ; Rev ) φR φRev  RC(λt ; Revt ) oU ooo o o oo ev  ooo θR RC(λt ; Rt ) σp σ1 G CLR(λ; sp R) ll l ev lll l lll vlll G CLR(λ; s1 (Rev )) φs1 (Rev ) φsp R  RC(λt ; (s1 Revt )) h    θsevp R   RC(λt ; sp Rt ).

Since m −1 (ν (k−1) ) = 1, Case 2 is also impossible. So Case 1 holds for ν (k−1) . It follows that m (k−1) −1 (ν (k) ) = 0, that is, m −2 (ν (k) ) = 0. But this is a contradiction. Suppose that m −1 (ν (k−1) ) = 0. Then (k−1) ≤ − 2 and (k−1) (k) ≤ − 2. This yields a contradiction unless > 2. By the minimality of and (k) , (k) of length − 2 that are either singular or have there cannot be strings in ν (k) label zero, so it follows that either m −2 (ν (k) ) = 0 or P −2 (ν) ≥ 2. 7) the latter immediately yields the desired result m −1 (ν (k+1) ) = 0, so assume that (k) m −2 (ν (k) ) = 0 and P −2 (ν) ≤ 1.

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


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