- Schur polynomial
In
mathematics , Schur polynomials, named afterIssai Schur , are certainsymmetric polynomial s in "n" variables with integral coefficients. Theelementary symmetric polynomial s and thecomplete homogeneous symmetric polynomial s are special cases of Schur polynomials. Schur polynomials depend on a partition of natural number "d" into at most "n" parts, and are then homogeneous of degree "d". The importance of Schur polynomials stems for a large part from their role inrepresentation theory , notably ofsymmetric group s or ofgeneral linear group s, where they correspond toirreducible representation s. The set of all Schur polynomials for fixed values of "d" form a linear basis for the space of all symmetric polynomials that are homogeneous of degree "d". As a consequence every symmetric polynomial "P" can be written in a unique way as a linear combination of Schur polynomials; moreover if "P" has integral coefficients, then the linear combination has so as well. 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 theLittlewood-Richardson rule .Definition
Schur polynomials correspond to
integer partition s. Given a partition:
(where each is a non-negative integer), we can compute the corresponding Schur polynomial by expanding determinants
:
This gives a symmetric "function" because the numerator and denominator are each determinants which change sign under any transposition of the variables. Furthermore, the denominator is a
Vandermonde determinant ::
Each factor divides the determinant in the numerator, so the quotient is a polynomial, which can moreover be seen to be homogeneous of degree .
Properties
Because we can readily enumerate the distinct partitions of "d" into "n" parts using
Ferrers diagram s, using this formula we can write down all the degree "d" Schur polynomials in "n" variables, giving a linear basis for the space of homogeneous degree "d" symmetric polynomials in "n" variables.Each Schur polynomial in "n" variables is a polynomial function of the
elementary symmetric polynomial s:::
and so forth, down to
:
Explicit expressions can be found using computational techniques from
elimination theory , perhaps the most elementary of which are Gröbner bases using anelimination order .For a partition , the Schur function can be expanded as a sum of monomials as
:
where the summation is over all semistandard
Young tableau x of shape ; the exponents give the weight of , in other words each counts the occurrences of the number in .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 forth. 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 thesymmetric group s,general linear group s, andunitary group s, and in fact this is how they arose. TheWeyl character formula helps to generalize Schur's work to other compact and semisimpleLie group s.ee also
*
symmetric polynomial
*elementary symmetric polynomial
*Galois theory
*Issai Schur References
*springer|id=s/s120040|title=Schur functions in algebraic combinatorics|first=Bruce E. |last=Sagan
*cite book | author=Sturmfels, Bernd | title=Algorithms in Invariant Theory | location=New York | publisher=Springer | year=1993 | id=ISBN 0-387-82445-6 is a beautiful introduction to computational methods in invariant theory.
*cite book | author=Tignol, Jean-Pierre | title=Galois's Theory of Algebraic Equations | location=Singapore | publisher=World Scientific | year=2001 | id=ISBN 981-02-4541-6 Offers some nice historical background.
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