A Simple Proof of the Hook Length Formula

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. In this note, we give a simple and direct proof for the "Hook Length Formula." The simplicity of our proof relies on the usage of the residue theorem as a short cut. The number of standard Young tableaux for a given Ferrers diagram. is the dimension of the irreducible representation of the symmetric group corresponding to X. The hook length formula (which was first proved in [1]) is a method for calculating this number and is a surprisingly beautiful formula (because the problem looks complex but the formula looks naive). Let us begin by giving the precise definitions of Ferrers diagrams, standard Young tableaux, and the hook length formula. Suppose that N and m are positive integers and that k = (kI, ..., Xm) is a sequence in N U {0) such that Xk > k2 > "'" > X-m and EmI, i = N. We can think of this as an array of boxes in which the number of boxes in the top row is i , the number of boxes in the second row is k2, and so on. Such a diagram is called a Ferrers diagram. An interesting question is: How many standard Young tableaux are therefor a given Ferrers diagram? As noted, the hook length formula is an answer to this question. right of that node (that's why these sums are called hook lengths).