?W H A T I S . . .a Young Tableau?

Alexander Yong

Young tableaux are ubiquitous combinatorial ob-

jectsmaking important and inspiring appearancesin representation theory, geometry, and algebra.

They naturally arise in the study of symmetric

functions, representation theory of the symmetric

and complex general linear groups, and Schu-

bert calculus of Grassmannians. Discovering andinterpreting enumerative formulas for Young

tableaux (and their generalizations) is a core theme

ofalgebraic combinatorics.

Let = (1 2 . . . k 0) be a partition

of size || = 1 + . . .+ k, identifiedwith itsYoung

diagram: a left-justified shapeofk rows of boxesoflength1, . . . , k. Forexample, = (4,2,1) isdrawn

. A (Young) filling of assigns a positive

integer to each box of , e.g., 2 1 1 46 24

. A filling is

semistandard if the entries weakly increase along

rows and strictly increase along columns. A semi-

standardfilling is standard if it is a bijective assign-

ment of {1,2 . . . , ||}. So 1 2 2 42 34

is a semistan-

dard Young tableau while 1 3 4 62 75

is a standard

Young tableau, bothofshape.

Alexander Yong is Dunham Jackson Assistant Professor

at the University of Minnesota, Minneapolis, and a visit-

ing member at the Fields Institute, Toronto, Canada. His

email address is ayong@math.umn.edu.

The author thanks Sergey Fomin, Victor Reiner, Hugh

Thomas, Alexander Woo, and the editors for helpful

suggestions. The author was partially supported by

NSF grant DMS 0601010 and an NSERC postdoctoral

fellowship.

We focus on the enumeration and generating

series of Young tableaux. Frame-Robinson-Thralls

elegant (and nontrivial) hook-length formula

states that the number of standardYoung tableaux

of shape is f :=||!b hb

, where the product in the

denominator is over all boxes b of and hb is the

hook-length of b, i.e., the number of boxes directly

to the right or below b (including b itself). Thus,

f (4,2,1) =7!

6 4 2 1 3 1 1= 35.

A similar hook-content formula counts the

number of semistandard Young tableaux, but we

now consider instead their generating series: fix

and a bound N on the size of the entries in each

semistandard tableau T . Let xT =N

i=1 x#is in Ti .

The Schur polynomial is the generating series

s(x1, . . . , xN) :=semistandard T x

T . For exam-

ple, when N = 3 and = (2,1) there are eight

semistandardYoung tableaux:

1 12

, 1 22

, 1 32

, 1 23

, 1 13

, 1 33

, 2 23

, 2 33

.

The corresponding Schur polynomial, with termsin the same order, is s(2,1)(x1, x2, x3) = x

21x2 +

x1x22+ x1x2x3+ x1x2x3+ x

21x3 + x1x

23 + x

22x3+ x2x

23.

In general, these are symmetric polynomials, i.e.,

s(x1, . . . , xN) = s(x(N), . . . , x(N)) for all in the

symmetric group SN (the proof is a clever trick

knownas theBender-Knuth involution).

Both the irreducible complex representations of

Sn and the irreducible degree n polynomial repre-

sentations of the general linear group GLN(C) are

indexed by partitions with || = n. The associ-

ated irreducibleSn-representation has dimension

equal to f , while the irreducibleGLN(C) represen-

tation has character s(x1, . . . , xN). These facts can

240 Notices of the AMS Volume 54, Number 2

be proved with an explicit construction of the re-spective representations having a basis indexed bytheappropriate tableaux.

In algebraic geometry, the Schubert varieties,in the complex Grassmannian manifold Gr(k,Cn)of k-planes in Cn, are indexed by partitions con-tained inside ak(nk) rectangle.Here, the Schurpolynomial s(x1, . . . , xk) represents the class ofthe Schubert variety under a natural presentationof thecohomologyringH(Gr(k,Cn)).

Schur polynomials form a vector space basis(say, overQ) of the ring of symmetric polynomialsin the variables x1, . . . , xN . Since a product of sym-metric polynomials is symmetric, we can expandthe result in termsof Schurpolynomials. Inparticu-lar, define the Littlewood-Richardson coefficientsC, by

(1)

s(x1, . . . , xN) s(x1, . . . , xN) =

C, s(x1, . . . , xN).

In fact, C, Z0! These numbers count tensor

product multiplicities of irreducible representa-tions ofGLN(C). Alternatively, they count Schubertcalculus intersection numbers for a triple of Schu-bert varieties in a Grassmannian. However, neitherof these descriptions of C, is really a means to

calculate thenumber.The Littlewood-Richardson rule combinatori-

ally manifests the positivity of the C, . Numerous

versions of this rule exist, exhibiting different fea-tures of the numbers. Here is a standard version:take theYoungdiagramof and remove the Youngdiagram of , where the latter is top left justifiedin the former (if is not contained inside , thendeclare C, = 0); this skew-shape is denoted /.

Then C, counts the number of semistandard

fillings T of shape / such that (a) as the entriesare read along rows from right to left, and fromtop to bottom, at every point, the number of isappearing always is weakly less than the numberof i 1s, for i 2; and (b) the total number of is

appearing is i. Thus C(3,2,1)(2,1),(2,1) = 2 is witnessed

by

11

2

and 12

1

.

Extending the GLN(C) story, a generalizedLittlewood-Richardson rule exists for all com-plex semisimple Lie groups, where Littelmannpaths generalize Young tableaux. In contrast, thesituation is much less satisfactory in the Schu-bert calculus context, although in recent workwith Hugh Thomas, we made progress for the(co)minusculegeneralizationofGrassmannians.

No discussion of Young tableaux is completewithout the Schensted correspondence. Thisassociates each Sn bijectively with pairsof standard Young tableaux (T , U) of the same

shape , where || = n. This can be used to provethe Littlewood-Richardson rule but is noteworthyin its own right in geometry and representationtheory.

Given a permutation (in one-line notation), e.g., = 21453 S5, at each step i we add a box into

some row of the current insertion tableau T : ini-tially insert(i) into the first rowof T . If no entriesy of that row are larger than (i), place (i) in anew box at the end of the row and place a new boxcontaining i at thesameplace in thecurrent record-

ing tableau U . Otherwise, let (i) replace the left-most y > (i) and insert y into the second row,and so on. This eventually results in two tableauxof the same shape; Schensted outputs (T , U) afternsteps. Inourexample, thestepsare

(,), ( 2 , 1 ),

(12, 12

),

(1 42

, 1 32

),

(1 4 52

, 1 3 42

),

(1 3 52 4

, 1 3 42 5

)= (T , U).

It is straightforward to prove well-definedness andbijectivity of this procedure. Also, T and U encodeinteresting information about . For example, itis easy to show that 1 equals the length of thelongest increasing subsequence in (see e.g. workofBaik-Deift-Johansson for connections to randommatrix theory). A sample harder fact is that if corresponds to (T , U) then 1 corresponds to(U, T).

An excellent source for more on the combi-natorics of Young tableaux is [Sta99], whereasapplications to geometry and representation theo-ry are developed in [Ful97]. For a survey containingexamples of Young tableaux for other Lie groups,see [Sag90]. Active research on the topic of Youngtableaux continues. For example, recently in col-laboration with Allen Knutson and Ezra Miller, wefound a simplicial ball of semistandard tableaux,together with applications to Hilbert series formu-laeofdeterminantal ideals.

Further Reading[Ful97] W. Fulton, Young tableaux. With applications

to representation theory and geometry, London

Mathematical Society Student Texts 35 (1997),

Cambridge University Press, Cambridge.

[Sag90] B. Sagan, The ubiquitous Young tableau, In-

variant theory and tableaux (Minneapolis, MN,

1988), 262298, 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

appendix 1 by Sergey Fomin, Cambridge Studies

in Advanced Mathematics 62 (1999), Cambridge

University Press, Cambridge.

February 2007 Notices of the AMS 241