Skew Young tableau

Consider two partitions, of and of . With those partitions one can associate Young diagrams, or Ferrers diagrams (cf. Young diagram), also denoted by and . Suppose that each cell of is also a cell of . The set-difference contains exactly cells. It is called a skew Young diagram, or skew Ferrers diagram.

Such a diagram can be filled with integers from to in increasing order in each row and in each column. This is called a standard skew Young tableau. If repetitions are allowed and if the rows are only non-decreasing, the tableau is called semi-standard. These are generalizations of Young tableaux (cf. Young tableau).

For example,

Figure: s110150a

are standard, respectively semi-standard, of shape .

It is possible to define the Schur function combinatorially as the generating function of all semi-standard Young tableaux of shape filled with indices of elements of , as follows.

Let be an alphabet. Let be a semi-standard Young tableau of shape , containing entries from to . The product of the indeterminates whose indices appear in is a monomial. Then is the sum of those monomials over the set of all possible such tableaux .

When replacing Young tableaux by skew Young tableaux of shape , one obtains the skew Schur function . Those functions have many properties in common with ordinary Schur functions. See [a3] for Schur functions.

This connection between combinatorial (Young tableaux) and algebraic (Schur functions) objects is very fruitful, both for combinatorics and for algebra. For applications to -analysis, cf., e.g., [a1] and [a2].

References

[a1]	J. Désarménien, "Fonctions symétriques associées à des suites classiques de nombres" Ann. Sci. Ecole Normale Sup. , 16 (1983) pp. 231–304
[a2]	J. Désarménien, D. Foata, "Fonctions symétriques et séries hypergéométriques basiques multivariées" Bull. Soc. Math. France , 113 (1985) pp. 3–22
[a3]	I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1995) (Edition: Second)