Schur polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur functions can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials.
Definition
Schur polynomials are indexed by integer partitions. Given a partition λ = (λ_{1}, λ_{2}, …,λ_{n}), where λ_{1} ≥ λ_{2}≥ … ≥ λ_{n}, and each λ_{j} is a non-negative integer, the functions
are alternating polynomials by properties of the determinant. A polynomial is alternating if it changes sign under any transposition of the variables.
Since they are alternating, they are all divisible by the Vandermonde determinant,
The Schur polynomials are defined as the ratio
This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the Vandermonde determinant.
Properties
The degree d Schur polynomials in n variables are a linear basis for the space of homogeneous degree d symmetric polynomials in n variables. For a partition λ = (λ_{1}, λ_{2}, ..., λ_{n}), the Schur polynomial is a sum of monomials,
where the summation is over all semistandard Young tableaux T of shape λ. The exponents t_{1}, ..., t_{n} give the weight of T, in other words each t_{i} counts the occurrences of the number i in T. This can be shown to be equivalent to the definition from the first Giambelli formula using the Lindström–Gessel–Viennot lemma (as outlined on that page).
Schur polynomials can be expressed as linear combinations of monomial symmetric functions m_{μ} with non-negative integer coefficients K_{λμ} called Kostka numbers,
The Kostka numbers K_{λμ} are given by the number of semi-standard Young tableaux of shape λ and weight μ.
Jacobi−Trudi identities
The first Jacobi−Trudi formula expresses the Schur polynomial as a determinant in terms of the complete homogeneous symmetric polynomials,
- ^{[1]}
where h_{i} := s_{(i)}.
The second Jacobi-Trudi formula expresses the Schur polynomial as a determinant in terms of the elementary symmetric polynomials,
- ^{[2]}
where e_{i} := s_{(1i)}. and λ' is the conjugate partition to λ.
These two formulae are known as determinantal identities.
The Giambelli identity
Another determinantal identity is Giambelli's formula, which expresses the Schur function for an arbitrary partition in terms of those for the hook partitions contained within the Young diagram. In Frobenius' notation, the partition is denoted
where, for each diagonal element in position ii, a_{i} denotes the number of boxes to the right in the same row and b_{i} denotes the number of boxes beneath it in the same column (the arm and leg lengths, respectively).
The Giambelli identity expresses the partition as the determinant
- .
The Cauchy identity
The Cauchy identities for the Schur functions (now in infinitely many variables), states that
and
where the sum is taken over all partitions λ. There are many generalizations of these identities, for example, Hall-Littlewood polynomials, Schubert polynomials and Grothendieck polynomials admit Cauchy-like identities.
The Murnaghan−Nakayama rule
The Murnaghan–Nakayama rule expresses a product of a power-sum symmetric function with a Schur polynomial, in terms of Schur polynomials:
where the sum is over all partitions μ such that μ/λ is a rim-hook of size r and ht(μ/λ) is the number of rows in the diagram μ/λ.
The Littlewood-Richardson rule and Pieri's formula
The Littlewood–Richardson coefficients depend on three partitions, say , of which and describe the Schur functions being multiplied, and gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients such that
The Littlewood–Richardson rule states that is equal to the number of Littlewood–Richardson tableaux of skew shape and of weight .
Pieri's formula is a special case of the Littlewood-Richardson rule, which expresses the product in terms of Schur polynomials. The dual version expresses in terms of Schur polynomials.
Specializations
Evaluating the Schur polynomial s_{λ} in (1,1,...,1) gives the number of semi-standard Young tableaux of shape λ with entries in 1, 2, ..., n. One can show, by using the Weyl character formula for example, that
In this formula, λ, the tuple indicating the width of each row of the Young diagram, is implicitly extended with zeros until it has length n. The sum of the elements λ_{i} is d. See also the Hook length formula which computes the same quantity for fixed λ.
Example
The following extended example should help clarify these ideas. Consider the case n = 3, d = 4. Using Ferrers diagrams or some other method, we find that there are just four partitions of 4 into at most three parts. We have
and so on. Summarizing:
Every homogeneous degree-four symmetric polynomial in three variables can be expressed as a unique linear combination of these four Schur polynomials, and this combination can again be found using a Gröbner basis for an appropriate elimination order. For example,
is obviously a symmetric polynomial which is homogeneous of degree four, and we have
Relation to representation theory
The Schur polynomials occur in the representation theory of the symmetric groups, general linear groups, and unitary groups. The Weyl character formula implies that the Schur polynomials are the characters of finite-dimensional irreducible representations of the general linear groups, and helps to generalize Schur's work to other compact and semisimple Lie groups.
Several expressions arise for this relation, one of the most important being the expansion of the Schur functions s_{λ} in terms of the symmetric power functions . If we write χλ
ρ for the character of the representation of the symmetric group indexed by the partition λ evaluated at elements of cycle type indexed by the partition ρ, then
where ρ = (1^{r1}, 2^{r2}, 3^{r3}, ...) means that the partition ρ has r_{k} parts of length k.
A proof of this can be found in R. Stanley's Enumerative combinatoric II, Corollary 7.17.5.
The integers χλ
ρ can be computed using the Murnaghan–Nakayama rule.
Skew Schur functions
Skew Schur functions s_{λ/μ} depend on two partitions λ and μ, and can be defined by the property
Here, the inner product is the Hall inner product, for which the Schur polynomials form an orthonormal basis.
Similar to the ordinary Schur polynomials, there are numerous ways to compute these. The corresponding Jacobi-Trudi identities are
- ,
- .
There is also a combinatorial interpretation of the skew Schur polynomials, namely it is a sum over all semi-standard Young tableaux (or column-strict tableaux) of the skew shape .
The skew Schur polynomials expands positively in Schur polynomials. A rule for the coefficients is given by the Littlewood-Richardson rule.
Generalizations
There are numerous generalizations of Schur polynomials:
- Hall–Littlewood polynomials
- Shifted Schur polynomials
- Factorial Schur polynomials
- Flagged Schur polynomials
- Double Schur polynomials
- Schubert polynomials
- Stanley symmetric functions (also known as stable Schubert polynomials)
- Key polynomials (also known as Demazure characters)
- Quasi-symmetric Schur polynomials
- Jack polynomials
- Modular Schur polynomials
- Macdonald polynomials
- Schur polynomials for the symplectic and orthogonal group.
- k-Schur functions
- Loop Schur functions
- Grothendieck polynomials (K-theoretical analogue of Schur polynomials)
- LLT polynomials
Double Schur polynomials
The double Schur polynomials^{[3]} can be seen as a generalization of the shifted Schur polynomials. These polynomials are also closely related to the factorial Schur polynomials. Given a partition λ, and a sequence a_{1}, a_{2},… one can define the double Schur polynomial s_{λ}(x || a) as
where the sum is taken over all reverse semi-standard Young tableaux T of shape λ, and integer entries in 1,…,n. Here T(α) denotes the value in the box α in T and c(α) is the content of the box.
A combinatorial rule for the Littlewood-Richardson coefficients (depending on the sequence a), is given by A.I Molev in.^{[3]} In particular, this implies that the shifted Schur polynomials have non-negative Littlewood-Richardson coefficients.
The shifted Schur polynomials, s^{*}_{λ}(y) , can be obtained from the double Schur polynomials by specializing a_{i}=-i and y_{i}=x_{i}+i.
The double Schur polynomials are special cases of the double Schubert polynomials.
Factorial Schur polynomials
The factorial Schur polynomials may be defined as follows. Given a partition λ, and a doubly infinite sequence …,a_{-1}, a_{0}, a_{1}, … one can define the factorial Schur polynomial s_{λ}(x|a) as
where the sum is taken over all semi-standard Young tableaux T of shape λ, and integer entries in 1,…,n. Here T(α) denotes the value in the box α in T and c(α) is the content of the box.
There is also a determinant formula,
where (y|a)^{k} = (y-a_{1})... (y-a_{k}). It is clear that if we let a_{i}=0 for all i, we recover the usual Schur polynomial s_{λ}.
The double Schur polynomials and the factorial Schur polynomials in n variables are related via the identity s_{λ}(x||a) = s_{λ}(x|u) where a_{n-i+1} = u_{i}.
See also
- Schur functor
- Littlewood–Richardson rule, where one finds some identities involving Schur polynomials.
- Schubert polynomials, a generalization of Schur polynomials.
References
- Macdonald, I. G. (1995). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (2nd ed.). The Clarendon Press Oxford University Press. ISBN 978-0-19-853489-1. MR 1354144.
- Sagan, Bruce E. (2001), "Schur functions in algebraic combinatorics", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Sturmfels, Bernd (1993). Algorithms in Invariant Theory. New York: Springer. ISBN 0-387-82445-6.
- ↑ Formula A.5 in Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249, ISBN 978-0-387-97527-6.
- ↑ Formula A.6 in Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249, ISBN 978-0-387-97527-6.
- 1 2 Molev, A.I. (June 2009). "Littlewood–Richardson polynomials". Journal of Algebra. 321 (11): 3450–3468. doi:10.1016/j.jalgebra.2008.02.034.