The Ubiquitous Young Tableau

Young tableaux have found extensive application in combinatorics [Vie 84], group representations [Jam 78], invariant theory [DRS 74, DKR 78], symmetric funcions [Mad 79], and the theory of algorithms [Knu 73, pages 48-72]. This paper is an expository treatment of some of the highlights of tableaux theory. These include the hook and determinantal formulae for enumeration of both standard and generalized tableaux, their connection with irreducible representations of matrix groups, and the Robinson-Schensted-Knuth algorithm. 1 Three families of tableaux Young tableaux were first introduced in 1901 by the Reverend Alfred Young [You 01, page 133] as a tool for invariant theory. Subsequently, he showed that they can give information about representations of symmetric groups. Since then, tableaux have played an important role in many areas of mathematics from enumerative combinatorics to algebraic geometry. This paper is a survey of some of these applications. In recent years the number of tableaux of various types has been increasing at an impressive rate. To limit this paper to a reasonable length, our discussion will be restricted to three fundamental families of tableaux: ordinary, shifted and oscillating. The rest of this section will be devoted to the definitions and notation need to describe these arrays. In Section 2 we present the hook and determinantal formulae for enumeration of standard tableaux. The third section examines the connection with representations of the symmetric group. The Robinson-Schensted algorithm appears in Section 4 as a combinatorial way of explaining the decomposition of the regular representation. The next four sections rework the material from the first four using generalized tableaux (those with repeated entries), representations of general linear and symplectic groups, and the theory of symmetric functions. Section 9 is a brief exposition of some open problems. 1.1 Ordinary tableaux In what follows, N and P stand for the non-negative and positive integers respectively. A partition λ of n ∈ N, written λ ` n, is a sequence of positive integers λ = (λ1, λ2, · · · , λl) in weakly decreasing order such that ∑l i=1 λi = n. The λi are called the parts of λ. The unique partition of 0 is λ = φ. The shape of λ is an array of boxes (or dots or cells) with l left-justified rows and λi boxes in row i. We will use λ to represent both the partition and its shape, while (i, j) will denote the cell in row i and column j. By way of illustration, the following figure shows the shape of the partition λ = (2, 2, 1) ` 5 with cell (3,1) displayed as a diamond. In deference to Alfred Young’s nationality, we have chosen to draw partition shapes in the English style, i.e., as if they were part of a matrix. The reader should be aware that some mathematicians (notably the French) prefer to use the conventions of coordinate geometry where λ1 cells are placed along the x-axis, λ2 cells are placed along the line y = 1, etc. To them and to Rene Descartes, we abjectly apologize.