Skew Young tableau

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

Such a diagram can be filled with integers from $1$ to $n$ 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,

     ┌───┬───┬───┐      ┌───┬───┬───┐ 
     │ 1 │ 2 │ 7 │      │ 1 │ 1 │ 3 │ 
     ├───┼───┼───┘      ├───┼───┼───┘ 
     │ 3 │ 6 │          │ 2 │ 2 │     
 ┌───┼───┼───┘      ┌───┼───┼───┘     
 │ 4 │ 5 │          │ 1 │ 3 │         
 └───┴───┘    and   └───┴───┘        

are standard, respectively semi-standard, of shape $(4,3,2) / (1,1)$.

It is possible to define the Schur function $S_\lambda(X)$ combinatorially as the generating function of all semi-standard Young tableaux of shape $\lambda$ filled with indices of elements of $X$, as follows.

Let $X=\{x_1,\ldots,x_k\}$ be an alphabet. Let $T$ be a semi-standard Young tableau of shape $\lambda$, containing entries from $1$ to $k$. The product of the indeterminates whose indices appear in $T$ is a monomial. Then $S_\lambda(X)$ is the sum of those monomials over the set of all possible such tableaux $T$.

When replacing Young tableaux by skew Young tableaux of shape $\lambda/\mu$, one obtains the skew Schur function $S_{\lambda/\mu}(X)$. 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 $q$-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)