WHAT IS... ? a Young Tableau? Alexander Yong

Young tableaux are ubiquitous combinatorial ob- We focus on the enumeration and generating jects making important and inspiring appearances series of Young tableaux. Frame-Robinson-Thrall’s in representation theory, geometry, and algebra. elegant (and nontrivial) hook-length formula They naturally arise in the study of symmetric states that the number of standard Young tableaux functions, representation theory of the symmetric |λ|! of shape λ is f λ := , where the product in the and complex general linear groups, and Schu- b hb bert calculus of Grassmannians. Discovering and denominator is over all boxes b of λ and h is the Q b interpreting enumerative formulas for Young hook-length of b, i.e., the number of boxes directly tableaux (and their generalizations) is a core theme to the right or below b (including b itself). Thus, of algebraic combinatorics. 7! f (4,2,1) = = 35. Let λ = (λ1 ≥ λ2 ≥ ... ≥ λk ≥ 0) be a partition 6 · 4 · 2 · 1 · 3 · 1 · 1 of size |λ| = λ1 + ... + λk, identified with its Young A similar hook-content formula counts the diagram: a left-justified shape of k rows ofboxesof number of semistandard Young tableaux, but we λ ,...,λ λ = ( , , ) length 1 k.Forexample, 4 2 1 isdrawn now consider instead their generating series: fix λ .A (Young) filling of λ assigns a positive and a bound N on the size of the entries in each T N #i’s in T semistandard tableau T . Let x = i=1 xi . 2 1 1 4 The Schur polynomial is the generatingQ series integer to each box of λ, e.g., . A filling is T 6 2 sλ(x1,...,xN ) := semistandard T x . For exam- 4 ple, when N = 3 andP λ = (2, 1) there are eight semistandard if the entries weakly increase along semistandard Young tableaux: rows and strictly increase along columns. A semi- standard filling is standard ifitisabijectiveassign- 1 1 , 1 2 , 1 3 , 1 2 , 1 1 , 1 3 , 2 2 , 2 3 . 2 2 2 3 3 3 3 3 ment of {1, 2 ..., |λ|}. So 1 2 2 4 is a semistan- 2 3 The corresponding Schur polynomial, with terms 4 in the same order, is s (x , x , x ) = x2x + 1 3 4 6 (2,1) 1 2 3 1 2 dard Young tableau while is a standard 2 + + + 2 + 2 + 2 + 2 2 7 x1x2 x1x2x3 x1x2x3 x1x3 x1x3 x2x3 x2x3. 5 In general, these are symmetric polynomials, i.e., Young tableau,bothof shape λ. sλ(x1,...,xN ) = sλ(xσ(N),...,xσ(N)) for all σ in the symmetric group SN (the proof is a “clever trick” Alexander Yong is Dunham Jackson Assistant Professor known as the Bender-Knuth involution). at the University of Minnesota, Minneapolis, and a visit- Both the irreducible complex representations of ing member at the Fields Institute, Toronto, Canada. His Sn and the irreducible degree n polynomial repre- email address is [email protected]. sentations of the general linear group GLN (C) are The author thanks Sergey Fomin, Victor Reiner, Hugh indexed by partitions λ with |λ| = n. The associ- Thomas, Alexander Woo, and the editors for helpful S suggestions. The author was partially supported by ated irreducible n-representation has dimension λ NSF grant DMS 0601010 and an NSERC postdoctoral equal to f , while the irreducible GLN (C) represen- fellowship. tation has character sλ(x1,...,xN ). These facts can

240 Notices of the AMS Volume 54, Number 2 be proved with an explicit construction of the re- shape λ, where |λ| = n. This can be used to prove spective representations having a basis indexed by the Littlewood-Richardson rule but is noteworthy the appropriate tableaux. in its own right in geometry and representation In algebraic geometry, the Schubert varieties, theory. in the complex Grassmannian manifold Gr(k, Cn) Given a permutation (in one-line notation), e.g., n of k-planes in C , are indexed by partitions λ con- σ = 21453 ∈ S5, at each step i we add a box into tained inside a k×(n −k) rectangle. Here, the Schur some row of the current insertion tableau T : ini- polynomial sλ(x1,...,xk) represents the class of tially insert σ (i) into the first row of T . If no entries the Schubert variety under a natural presentation y of that row are larger than σ (i), place σ (i)e in a ofthecohomologyring H⋆(Gr(k, Cn)). new box at the end of the row and placee a new box Schur polynomials form a vector space basis containing i at the same place in the current record- (say, over Q) of the ring of symmetric polynomials ing tableau U. Otherwise, let σ (i) replace the left- in the variables x1,...,xN . Since a product of sym- most y > σ (i) and insert y into the second row, metric polynomials is symmetric, we can expand and so on. Thise eventually results in two tableaux the result in terms of Schur polynomials. In particu- of the same shape; Schensted outputs (T , U) after lar, define the Littlewood-Richardson coefficients n steps.Inourexample,thestepsare ν Cλ,µ by s (x ,...,x ) · s (x ,...,x ) = λ 1 N µ 1 N (∅, ∅), ( 2 , 1 ), 1 , 1 , 1 4 , 1 3 , (1) ν 2 2 ! 2 2 ! Cλ,µ sν (x1,...,xN ). Xν ν In fact, Cλ,µ ∈ Z≥0! These numbers count tensor 1 4 5 , 1 3 4 , 1 3 5 , 1 3 4 = (T , U). product multiplicities of irreducible representa- 2 2 ! 2 4 2 5 ! tions of GLN (C). Alternatively, they count Schubert It is straightforward to prove well-definedness and calculus intersection numbers for a triple of Schu- bijectivity of this procedure. Also, T and U encode bert varieties in a Grassmannian. However, neither interesting information about σ . For example, it ν of these descriptions of Cλ,µ is really a means to is easy to show that λ1 equals the length of the calculate the number. longest increasing subsequence in σ (see e.g. work The Littlewood-Richardson rule combinatori- of Baik-Deift-Johansson for connections to random ν ally manifests the positivity of the Cλ,µ . Numerous matrix theory). A sample harder fact is that if σ versions of this rule exist, exhibiting different fea- corresponds to (T , U) then σ −1 corresponds to tures of the numbers. Here is a standard version: (U, T ). take the Young diagramof ν and removethe Young An excellent source for more on the combi- diagram of λ, where the latter is top left justified natorics of Young tableaux is [Sta99], whereas in the former (if λ is not contained inside ν, then applications to geometry and representation theo- ν ry are developed in [Ful97]. For a survey containing declare Cλ,µ = 0); this skew-shape is denoted ν/λ. ν examples of Young tableaux for other Lie groups, Then Cλ,µ counts the number of semistandard fillings T of shape ν/λ such that (a) as the entries see [Sag90]. Active research on the topic of Young are read along rows from right to left, and from tableaux continues. For example, recently in col- top to bottom, at every point, the number of i’s laboration with Allen Knutson and Ezra Miller, we appearing always is weakly less than the number found a simplicial ball of semistandard tableaux, of i − 1’s, for i ≥ 2; and (b) the total number of i′s together with applications to Hilbert series formu- (3,2,1) lae ofdeterminantalideals. appearing is µi. Thus C(2,1),(2,1) = 2 is witnessed by Further Reading 1 and 1 . 1 2 [Ful97] W. Fulton, Young tableaux. With applications 2 1 to representation theory and geometry, London Mathematical Society Student Texts 35 (1997), Extending the GLN (C) story, a generalized Cambridge University Press, Cambridge. Littlewood-Richardson rule exists for all com- [Sag90] B. Sagan, The ubiquitous Young tableau, In- plex semisimple Lie groups, where Littelmann variant theory and tableaux (Minneapolis, MN, paths generalize Young tableaux. In contrast, the 1988), 262–298, IMA Vol. Math. Appl., 19, situation is much less satisfactory in the Schu- Springer, New York, 1990. [Sta99] R. P. Stanley, Enumerative combinatorics. bert calculus context, although in recent work Vol. 2. With a foreword by Gian-Carlo Rota and with Hugh Thomas, we made progress for the appendix 1 by Sergey Fomin, Cambridge Studies (co)minuscule generalizationofGrassmannians. in Advanced Mathematics 62 (1999), Cambridge No discussion of Young tableaux is complete University Press, Cambridge. without the Schensted correspondence. This associates each σ ∈ Sn bijectively with pairs of standard Young tableaux (T , U) of the same

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