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

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Extra resources for A bijection between Littlewood-Richardson tableaux and rigged configurations

Example text

3, φrows(R) (iR (S)) = φR (S). By replacing R by rows(R) it may be assumed that R consists of single rows. The transpose maps are applied to reduce to the case of single columns. 7)]. For all S ∈ CLR(λ; R) we have cRt (trLR (S)) = n(R) − cR (S) by [24, Prop. 5]. 12)]. 1. From these facts, by replacing R with Rt it may be assumed that R consists of single columns. Applying the first reduction again, it may be assumed that R consists of single cells. Observe that in this case CLR(λ; R) = ST(λ). It is possible to verify cR (S) = cc(φR (S)) for S ∈ ST(λ) directly.

Let WR(R) (resp. WC(R)) be the set of standard words w such that the standard tableau P (w) is in RLR(λ; R) (resp. CLR(λ; R)). The relabeling bijection γR : RLR(λ; R) → CLR(λ; R) extends to a map WR(R) → WC(R) in an obvious way. This relabeling map satisfies P (γR (w)) = γR (P (w)) [24, Lemma 36]. It is not hard to see that # ◦ γR = γRev ◦ # as maps WR(R) → WC(Rev ). For all S ∈ RLR(λ; R) we have (γR (S))ev = P (#γR (S)) = P (#γR (word(S))) = P (γRev (#word(S))) = γRev (P (#word(S))) = γRev (S ev ).

Suppose first that L ≥ 2 and that RL has more than one column. Note that < R<∧ = R∧ . Vol. 8 (2002) Bijection between LR tableaux and rigged configurations CLR(λ; R) yyy yyyı∧ yyy yy9 CLR(λ; R∧ ) φR φR ∧ ı< ı< G CLR(λ; R< ) mm ı∧mmmm m m m vmmm G CLR(λ; R<∧ ) φR<∧  t RC(λt ; R∧ ) pU p pp p p pp ∧  ppp  RC(λt ; Rt ) 101 φR<  G RC(λt ; (R<∧ )t ) < h€€€  €€€ €€€ €€€ ∧  G RC(λt ; R< t ) <  The top face commutes since ı< and ı∧ are both relabelings that replace different subalphabets. The bottom face commutes since ∧ adds singular strings and < preserves colabels and hence preserves singularity of strings.

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