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symmetric function

Symmetric functions

Idea

A symmetric function is roughly a polynomial that is invariant under permutation of its variables. However, this is only strictly correct if the number of variables is finite, while symmetric functions depend on a countably infinite number of variables. The only symmetric polynomials in infinitely many variables are the constants. To fix this, one allows infinitely many terms, as long as the degree is finite.

For example, there is a 11-dimensional space of homogeneous symmetric functions of degree 11, with basis

x 1+x 2+= ix i. x_1 + x_2 + \cdots = \sum_i x_i .

There is a 22-dimensional space of homogeneous symmetric functions of degree 22, with basis

x 1 2+x 2 2+= ix i 2 x_1^2 + x_2^2 + \cdots = \sum_i x_i^2

and

i<jx ix j. \sum_{i \lt j} x_i x_j .

These basis elementa are called the elementary symmetric functions.

(The homogeneous symmetric functions of degree 00 are just the constants, as usual.)

There is also a noncommutative analogue: noncommutative symmetric functions.

Definitions

Let Λ n\Lambda_n be the ring consisting of polynomials in nn variables x 1,,x nx_1, \dots, x_n that are invariant under all permutations of the variables; these are the symmetric functions in nn variables or symmetric polynomials in nn variables. The rings Λ n\Lambda_n are graded by degree in the usual way, and there are homomorphisms of graded rings

Λ n+1Λ n, \Lambda_{n+1} \hookrightarrow \Lambda_{n} ,

given by setting the (n+1)(n+1)st variable equal to zero. Taking the limit (in the category-theoretic sense) of these rings Λ n\Lambda_n as nn \to \infty, we get the graded ring Λ \Lambda_\infty. This is usually called Λ\Lambda, or the ring of symmetric functions in countably many variables, or simply symmetric functions (or even symmetric polynomials, even though very few of them are really polynomials).

Alternatively, Λ\Lambda can be constructed as a colimit of rings using the homomorphisms

Λ nΛ n+1, \Lambda_{n} \hookrightarrow \Lambda_{n+1} ,

where we add in new terms with the new variable to make the result symmetric.

The definition depends on the ground field (or commutative ring or rig) kk, so we may write Λ(k)\Lambda(k) to be precise.

A query about the Hazewinkel’s description of the construction of Λ\Lambda as a (co)limit is here.

Properties

Symmetric functions play a fundamental role throughout representation theory, combinatorics and algebraic topology. The ring of symmetric functions, Λ\Lambda, has many interesting properties. For example, it is the free λ\lambda-ring on one generator (a coincidence of notation that has not been ignored). It is also a plethory. There are various bases of Λ\Lambda whose elements are in natural one-to-one correspondence with Young diagrams.

Λ\Lambda acquires many of these properties from the fact that it is the Grothendieck ring of the category of kk-linear species. This category is defined to be the functor category

[,Vect k] [\mathbb{P}, Vect_{k}]

where \mathbb{P} is the groupoid of finite sets and bijections, and Vect kVect_k is the category of vector spaces over a field kk of characteristic zero. The category of kk-linear species becomes a symmetric monoidal category thanks to Day convolution. It is also a semisimple abelian category, with a basis of objects given by irreducible representations of symmetric groups — one for each Young diagram. So, the Grothendieck group of this category becomes a commutative ring with a basis given by Young diagrams, and this is just Λ\Lambda.

Λ\Lambda is also the Grothendieck ring of a somewhat smaller SchurSchur, whose objects are Schur functors. There are various ways to think about these, but they can be identified with kk-linear species

F:Vect k F: \mathbb{P} \to Vect_{k}

with the special property that F(n)F(n) is finite-dimensional for all nn and 00 for nn sufficiently large. The category of Schur functors is again a semisimple abelian category with a basis of objects given by irreducible representations of symmetric groups, so its Grothendieck ring is again Λ\Lambda. For more on this, see Schur functor.

References

The classification described above of irreducible S nS_n-modules over \mathbb{C} also works unchanged for any algebraically closed field kk of characteristic zero. It also works when kk has characteristic pp and nn is not divisible by pp. There's an exercise at the end of section 6.1 of Serre's book (page 64 of the french edition) that says that if kk is a field of characteristic p>0p \gt 0 then the group algebra of the group GG is semisimple iff pp doesn't divide the order of GG.

Apart from this, the field matters a lot. There is a construction that gives all irreducible kS nk S_n-modules for any field kk, field, completely analogous to the Specht module construction over \mathbb{C}. However, it describes each module as a quotient module of a Specht module, and in general not even the dimension of these irreducible modules is known, let alone an explicit basis, or representing matrices.

The following article surveys the subject in light of the connections to Hopf algebras and also to noncommutative analogue:

  • Michiel Hazewinkel, Symmetric functions, noncommutative symmetric functions and quasisymmetric functions, pdf

Among the best books in the subject are

  • Gordon D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics 682, Springer, Berlin, 1978.
  • Gordon James, Adalbert Kerber, The representation theory of the symmetric group, With a foreword by P. M. Cohn. With an intr. by G. de B. Robinson. Enc. of Math. and its Appl. 16, Addison-Wesley 1981. xxviii+510 pp.

James also has a readable survey article that outlines developments in the ‘80’s and ‘90’s:

  • Gordon D. James, Symmetric groups and Schur algebras, in Algebraic Groups and their Representations, eds. R. W. Carter and J. Saxl, Kluwer Acad. Publ. Netherlands 1998.

Another approach is described here:

  • Alexander Kleshchev, Linear and Projective Representations of Symmetric Groups, Cambridge University Press, Cambridge 2009.

One can study symmetric functions in any characteristic, or over any integral domain. The power sum symmetric functions do not generate the ring of symmetric functions over \mathbb{Z}, and this difference matters. They appear to be of limited usefulness in the description of the modular irreducible representations of S nS_n.

Revised on March 29, 2014 03:26:58 by Urs Schreiber (185.37.147.12)