By Kirillov A.N., Schilling A., Shimozono M.
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Extra info for A bijection between Littlewood-Richardson tableaux and rigged configurations
Proof. Consider the diagram − G CLR(λ− ; R∧ ) CLR(λ; R) yyy nnT yyyı∧ − nnn n yyy n nn yy9 nnn ∧ CLR(λ; R ) φR φR ∧ φR∧ t RC(λt ; R∧ ) yyy pV ∧ yyy δ ppp p yyy p p yyy p 9 ppp G RC(λ−t ; R∧ t ). 6) which must be proved. The back left and back right faces commute by the definition of φ. The top triangle obviously commutes. It suffices to show that the bottom triangle commutes. This is done by computing δ ◦ ∧ explicitly. Let (ν, J) ∈ RC(λt ; Rt ). Then ∧ (ν, J) = (ν ∧ , J ∧ ) is obtained from (ν, J) by adding a singular string of length 1 to each of the first ηL − 1 rigged partitions.
2) we have cc(ν, J) − cc(ν, J) = α (1) (0) Sel. , New ser. (1) = α1 . The cocharge of an LR tableau T ∈ CLR(λ; R) is defined as coR (T ) = R − cR (T ) where R = i 3. Suppose the last rectangle is a single column. Consider the diagram: CLR(λ; R) xxx xxx− xxx xx9 CLR(λ− ; R) φR φR trLR G CLR(λt ; Rt ) nn − nnn n nn wnnn trLR G CLR(λ−t ; Rt ) φRt t RC(λ−t ; R ) U p pp p p ppp ppp δ RC(λt ; Rt ) trRC trRC φRt G RC(λ− ; R) g ∂ G RC(λ; R) Vol. 3. 6) with λ and R replaced by their transposes. 3 the front face commutes. 8. 3] embeddings were given between sets of LR tableaux of the form LRT(λ; R). 1)), thereby proving [13, Conjecture 18]. 1.
A bijection between Littlewood-Richardson tableaux and rigged configurations by Kirillov A.N., Schilling A., Shimozono M.
3. Suppose the last rectangle is a single column. Consider the diagram: CLR(λ; R) xxx xxx− xxx xx9 CLR(λ− ; R) φR φR trLR G CLR(λt ; Rt ) nn − nnn n nn wnnn trLR G CLR(λ−t ; Rt ) φRt t RC(λ−t ; R ) U p pp p p ppp ppp δ RC(λt ; Rt ) trRC trRC φRt G RC(λ− ; R) g ∂ G RC(λ; R) Vol. 3. 6) with λ and R replaced by their transposes. 3 the front face commutes. 8. 3] embeddings were given between sets of LR tableaux of the form LRT(λ; R). 1)), thereby proving [13, Conjecture 18]. 1.