[Math.CO] 2 Nov 2006

Home , Algebraic combinatorics, Schubert calculus, Schur polynomial, Young tableau

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ν (1) sλ(x1,...,xN) · sµ(x1,...,xN) = X Cλ,µ sν(x1,...,xN). ν ν In fact, Cλ,µ ∈ Z≥0! These numbers count tensor product multiplicities of irreducible representations of GLN(C). Alternatively, they count Schubert calculus intersection numbers for a triple of Schubert varieties in a Grassmannian. However, neither of these descrip- ν tions of Cλ,µ is really a means to calculate the number. ν The Littlewood-Richardson rule combinatorially manifests the positivity of the Cλ,µ. Numerous versions of this rule exist, exhibiting different features of the numbers. Here is a standard version: take the Young diagram of ν and remove the Young diagram of λ, where the latter is top left justiﬁed in the former (if λ is not contained inside ν, then declare ν ν Cλ,µ = 0); this skew-shape is denoted ν/λ. Then Cλ,µ counts the number of semistandard ﬁllings T of shape ν/λ such that (a) as the entries are read along rows from right to left, and from top to bottom, at every point, the number of i’s appearing always is weakly less ′ than the number of i − 1’s, for i ≥ 2; and (b) the total number of i s appearing is µi. Thus ( ) C 3,2,1 = 2 is witnessed by 1 and 1 . (2,1),(2,1) 1 2 2 1 Extending the GLN(C) story, a generalized Littlewood-Richardson rule exists for all complex semisimple Lie groups, where Littelmann paths generalize Young tableaux [Lit95]. In contrast, the situation is much less satisfactory in the Schubert calculus context, al- though in recent work with Hugh Thomas [ThoYon06], we made progress for the (co)minuscule generalization of Grassmannians. No discussion of Young tableaux is complete without the Schensted correspondence. This associates each σ ∈ Sn bijectively with pairs of standard Young tableaux (T, U) of the same shape λ, where |λ| = n. This can be used to prove the Littlewood-Richardson rule, but is noteworthy in its own right in geometry and representation theory.

Given a permutation (in one line notation), e.g., σ = 21453 ∈ S5, at each step i we add a box into some row of the current insertion tableau T: initially insert σ(i) into the ﬁrst T y σ(i) σ(i) row of . If no entries of that row are larger than e, place in a new box at the end of the row and place a new box containing i at the same place in the current recording e tableau U. Otherwise, let σ(i) replace the leftmost y > σ(i) and insert y into the second

e 2 row, and so on. This eventually results in two tableaux of the same shape; Schensted outputs (T, U) after n steps. In our example, the steps are

(∅, ∅), ( 2 , 1 ), 1 , 1 , 1 4 , 1 3 , 1 4 5 , 1 3 4 , 1 3 5 , 1 3 4 =(T, U). 2 2 2 2 2 2 2 4 2 5 It is straightforward to prove well-deﬁnedness and bijectivity of this procedure. Also, T and U encode interesting information about σ. For example, it is easy to show that λ1 equals the length of the longest increasing subsequence in σ (see e.g., work of Baik-Deift- Johansson [BaiDeiJoh99] for connections to random matrix theory). A sample harder fact is that if σ corresponds to (T, U) then σ−1 corresponds to (U, T). An excellent source for more on the combinatorics of Young tableaux is [Sta99], whereas applications to geometry and representation theory are developed in [Ful97]. For a sur- vey containing examples of Young tableaux for other Lie groups, see [Sag90]. Active research on the topic of Young tableaux continues, for example, recently in collaboration with Allen Knutson and Ezra Miller [KnuMilYon06], we found a simplicial ball of semistandard tableaux, together with applications to Hilbert series formulae of determinantal ideals.

REFERENCES [BaiDeiJoh99] J. Baik, P. Deift and K. Johansson, On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations, J. Amer. Math. Soc., 12 (1999), no.4, 1119–1178. [Ful97] W. Fulton, Young tableaux. With applications to representation theory and geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997. [KnuMilYon06] A. Knutson, E. Miller and A. Yong, Tableau complexes, Israel J. Math, to appear, 2006. [Lit95] P. Littelmann, Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499–525. [Sag90] B.Sagan, The ubiquitous Young tableau, Invariant theory and tableaux (Minneapolis, MN, 1988), 262–298, IMA Vol. Math. Appl., 19, Springer, New York, 1990. [Sta99] R. P. Stanley, Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and ap- pendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. [ThoYon06] H. Thomas and A. Yong, A combinatorial rule for (co)minuscule Schubert calculus, math.AG/0608276.

DEPARTMENT OF MATHEMATICS,UNIVERSITY OF MINNESOTA,MINNEAPOLIS, MN 55455, USA; AND THE FIELDS INSTITUTE, 222 COLLEGE STREET,TORONTO, ONTARIO, M5T 3J1, CANADA E-mail address: [email protected], [email protected]