Inmathematics, aYoung tableau (/tæˈbloʊ,ˈtæbloʊ/; plural:tableaux) is acombinatorial object useful inrepresentation theory andSchubert calculus. It provides a convenient way to describe thegroup representations of thesymmetric andgeneral linear groups and to study their properties.
Young tableaux were introduced byAlfred Young, amathematician atCambridge University, in 1900.[1][2] They were then applied to the study of the symmetric group byGeorg Frobenius in 1903. Their theory was further developed by many mathematicians, includingPercy MacMahon,W. V. D. Hodge,G. de B. Robinson,Gian-Carlo Rota,Alain Lascoux,Marcel-Paul Schützenberger andRichard P. Stanley.
Note: this article uses the English convention for displaying Young diagrams and tableaux.
AYoung diagram (also called aFerrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives apartitionλ of a non-negative integern, the total number of boxes of the diagram. The Young diagram is said to be of shapeλ, and it carries the same information as that partition. Containment of one Young diagram in another defines apartial ordering on the set of all partitions, which is in fact alattice structure, known asYoung's lattice. Listing the number of boxes of a Young diagram in each column gives another partition, theconjugate ortranspose partition ofλ; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.
There is almost universal agreement that in labeling boxes of Young diagrams by pairs of integers, the first index selects the row of the diagram, and the second index selects the box within the row. Nevertheless, two distinct conventions exist to display these diagrams, and consequently tableaux: the first places each row below the previous one, the second stacks each row on top of the previous one. Since the former convention is mainly used byAnglophones while the latter is often preferred byFrancophones, it is customary to refer to these conventions respectively as theEnglish notation and theFrench notation; for instance, in his book onsymmetric functions,Macdonald advises readers preferring the French convention to "read this book upside down in a mirror" (Macdonald 1979, p. 2). This nomenclature probably started out as jocular. The English notation corresponds to the one universally used for matrices, while the French notation is closer to the convention ofCartesian coordinates; however, French notation differs from that convention by placing the vertical coordinate first. The figure on the right shows, using the English notation, the Young diagram corresponding to the partition (5, 4, 1) of the number 10. The conjugate partition, measuring the column lengths, is (3, 2, 2, 2, 1).
In many applications, for example when definingJack functions, it is convenient to define thearm lengthaλ(s) of a boxs as the number of boxes to the right ofs in the diagram λ in English notation. Similarly, theleg lengthlλ(s) is the number of boxes belows. Thehook length of a boxs is the number of boxes to the right ofs or belows in English notation, including the boxs itself; in other words, the hook length isaλ(s) +lλ(s) + 1.
AYoung tableau is obtained by filling in the boxes of the Young diagram with symbols taken from somealphabet, which is usually required to be atotally ordered set. Originally that alphabet was a set of indexed variablesx1,x2,x3..., but now one usually uses a set of numbers for brevity. In their original application torepresentations of the symmetric group, Young tableaux haven distinct entries, arbitrarily assigned to boxes of the diagram. A tableau is calledstandard if the entries in each row and each column are increasing. The number of distinct standard Young tableaux onn entries is given by theinvolution numbers
In other applications, it is natural to allow the same number to appear more than once (or not at all) in a tableau. A tableau is calledsemistandard, orcolumn strict, if the entries weakly increase along each row and strictly increase down each column. Recording the number of times each number appears in a tableau gives a sequence known as theweight of the tableau. Thus the standard Young tableaux are precisely the semistandard tableaux of weight (1,1,...,1), which requires every integer up ton to occur exactly once.
In a standard Young tableau, the integer is adescent if appears in a row strictly below. The sum of the descents is called themajor index of the tableau.[3]
There are several variations of this definition: for example, in a row-strict tableau the entries strictly increase along the rows and weakly increase down the columns. Also, tableaux withdecreasing entries have been considered, notably, in the theory ofplane partitions. There are also generalizations such as domino tableaux or ribbon tableaux, in which several boxes may be grouped together before assigning entries to them.
Askew shape is a pair of partitions (λ,μ) such that the Young diagram ofλ contains the Young diagram ofμ; it is denoted byλ/μ. Ifλ = (λ1,λ2, ...) andμ = (μ1,μ2, ...), then the containment of diagrams means thatμi ≤ λi for alli. Theskew diagram of a skew shapeλ/μ is the set-theoretic difference of the Young diagrams ofλ andμ: the set of squares that belong to the diagram ofλ but not to that ofμ. Askew tableau of shapeλ/μ is obtained by filling the squares of the corresponding skew diagram; such a tableau is semistandard if entries increase weakly along each row, and increase strictly down each column, and it is standard if moreover all numbers from 1 to the number of squares of the skew diagram occur exactly once. While the map from partitions to their Young diagrams is injective, this is not the case for the map from skew shapes to skew diagrams;[4] therefore the shape of a skew diagram cannot always be determined from the set of filled squares only. Although many properties of skew tableaux only depend on the filled squares, some operations defined on them do require explicit knowledge ofλ andμ, so it is important that skew tableaux do record this information: two distinct skew tableaux may differ only in their shape, while they occupy the same set of squares, each filled with the same entries.[5] Young tableaux can be identified with skew tableaux in whichμ is the empty partition (0) (the unique partition of 0).
Any skew semistandard tableauT of shapeλ/μ with positive integer entries gives rise to a sequence of partitions (or Young diagrams), by starting withμ, and taking for the partitioni places further in the sequence the one whose diagram is obtained from that ofμ by adding all the boxes that contain a value ≤ i inT; this partition eventually becomes equal to λ. Any pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column; such shapes are calledhorizontal strips. This sequence of partitions completely determinesT, and it is in fact possible to define (skew) semistandard tableaux as such sequences, as is done by Macdonald (Macdonald 1979, p. 4). This definition incorporates the partitionsλ andμ in the data comprising the skew tableau.
Young tableaux have numerous applications incombinatorics,representation theory, andalgebraic geometry. Various ways of counting Young tableaux have been explored and lead to the definition of and identities forSchur functions.
Many combinatorial algorithms on tableaux are known, including Schützenberger'sjeu de taquin and theRobinson–Schensted–Knuth correspondence. Lascoux and Schützenberger studied anassociative product on the set of all semistandard Young tableaux, giving it the structure called theplactic monoid (French:le monoïde plaxique).
In representation theory, standard Young tableaux of sizek describe bases in irreducible representations of thesymmetric group onk letters. Thestandard monomial basis in a finite-dimensionalirreducible representation of thegeneral linear groupGLn are parametrized by the set of semistandard Young tableaux of a fixed shape over the alphabet {1, 2, ...,n}. This has important consequences forinvariant theory, starting from the work ofHodge on thehomogeneous coordinate ring of theGrassmannian and further explored byGian-Carlo Rota with collaborators,de Concini andProcesi, andEisenbud. TheLittlewood–Richardson rule describing (among other things) the decomposition oftensor products of irreducible representations ofGLn into irreducible components is formulated in terms of certain skew semistandard tableaux.
Applications to algebraic geometry center aroundSchubert calculus on Grassmannians andflag varieties. Certain importantcohomology classes can be represented bySchubert polynomials and described in terms of Young tableaux.
Young diagrams are in one-to-one correspondence withirreducible representations of thesymmetric group over thecomplex numbers. They provide a convenient way of specifying theYoung symmetrizers from which theirreducible representations are built. Many facts about a representation can be deduced from the corresponding diagram. Below, we describe two examples: determining the dimension of a representation and restricted representations. In both cases, we will see that some properties of a representation can be determined by using just its diagram. Young tableaux are involved in the use of the symmetric group inquantum chemistry studies of atoms, molecules and solids.[6][7]
Young diagrams also parametrize the irreducible polynomial representations of thegeneral linear groupGLn (when they have at mostn nonempty rows), or the irreducible representations of thespecial linear groupSLn (when they have at mostn − 1 nonempty rows), or the irreducible complex representations of thespecial unitary groupSUn (again when they have at mostn − 1 nonempty rows). In these cases semistandard tableaux with entries up ton play a central role, rather than standard tableaux; in particular it is the number of those tableaux that determines the dimension of the representation.
The dimension of the irreducible representationπλ of the symmetric groupSn corresponding to a partitionλ ofn is equal to the number of different standard Young tableaux that can be obtained from the diagram of the representation. This number can be calculated by thehook length formula.
Ahook lengthhook(x) of a boxx in Young diagramY(λ) of shapeλ is the number of boxes that are in the same row to the right of it plus those boxes in the same column below it, plus one (for the box itself). By the hook-length formula, the dimension of an irreducible representation isn! divided by the product of the hook lengths of all boxes in the diagram of the representation:
The figure on the right shows hook-lengths for all boxes in the diagram of the partition 10 = 5 + 4 + 1. Thus
Similarly, the dimension of the irreducible representationW(λ) ofGLr corresponding to the partitionλ ofn (with at mostr parts) is the number of semistandard Young tableaux of shapeλ (containing only the entries from 1 tor), which is given by the hook-length formula:
where the indexi gives the row andj the column of a box.[8] For instance, for the partition (5,4,1) we get as dimension of the corresponding irreducible representation ofGL7 (traversing the boxes by rows):
A representation of the symmetric group onn elements,Sn is also a representation of the symmetric group onn − 1 elements,Sn−1. However, an irreducible representation ofSn may not be irreducible forSn−1. Instead, it may be adirect sum of several representations that are irreducible forSn−1. These representations are then called the factors of therestricted representation (see alsoinduced representation).
The question of determining this decomposition of the restricted representation of a given irreducible representation ofSn, corresponding to a partitionλ ofn, is answered as follows. One forms the set of all Young diagrams that can be obtained from the diagram of shapeλ by removing just one box (which must be at the end both of its row and of its column); the restricted representation then decomposes as a direct sum of the irreducible representations ofSn−1 corresponding to those diagrams, each occurring exactly once in the sum.
Such arrangements were introduced by Alfred Young in 1900.