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

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And R< has at most two rectangles so that CLR(λ; R< ) is a singleton. Since φR t and φR< are bijections, RC(λt ; Rt ) and RC(λt ; R< ) are also singletons. So the < embedding  must send the unique element of RC(λt ; Rt ) to the unique element t of RC(λt ; R< ) and the required commutation follows. 2. 5) ❄ t t ✲ RC(λ ; R> ). >  Proof. 6 for R and R> , and θR > is injective. 3. 2. Removal of a cell from a single row Suppose RL is a single row. Then R∧ is given by splitting off one cell from the end of the row RL , and R∧ is obtained from R by removing the cell at the end t of RL .

If r > r a similar proof shows r = r and r = r. So it may be assumed (r) (r) (r) = ∞ implies that = = ∞ that r = r = r. Notice that (r) = so that r ≤ r and r ≤ r. Suppose Case 3 does not hold for (ν, J)(r−1) . Then (r−1) (r−1) (r−1) ≤ (r−1) < ∞ and ≤ < ∞ so that r ≥ r and r ≥ r. Thus r = r = r as desired. Otherwise suppose Case 3 holds for (ν, J)(r−1) . Recall that in Case 3 one has Otherwise (r−1) (r−1) = (r−1) λtr−1 = λtr . Suppose that (r−2) = (r−1) (r−1) . If < ∞, then r = r = r as desired.

Vol. 8 (2002) Bijection between LR tableaux and rigged configurations 115 In this section the following notation is used. Let R = (R1 , . . , RL ) be a sequence of rectangles with Ri = (ηiµi ), such that R1 and RL are single columns and |R| ≥ 2 and let λ be a partition and (ν, J) ∈ RC(λt ; Rt ). Write δ(ν, J) = (ν, J) δ(ν, J) = (ν, J) δ ◦ δ(ν, J) = (ν, J) δ ◦ δ(ν, J) = (ν, J). (k) (k) (k) , (k) , and denote the lengths of the strings that are Furthermore, let shortened in the transformations (ν, J) → (ν, J), (ν, J) → (ν, J), (ν, J) → (ν, J) and (ν, J) → (ν, J), respectively.

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