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symmetric function

Idea

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

For example, there is a $1$-dimensional space of homogeneous symmetric functions of degree $1$, with basis

$$x_1+x_2+\cdots =\sum _i{x}_i.$$x_1 + x_2 + \cdots = \sum_i x_i .

There is a $2$-dimensional space of homogeneous symmetric functions of degree $2$, with basis

$$x_1^2+x_2^2+\cdots =\sum _i{x}_i^2$$x_1^2 + x_2^2 + \cdots = \sum_i x_i^2

and

$$\sum _{i<j}{x}_i{x}_j.$$\sum_{i \lt j} x_i x_j .

(The homogeneous symmetric functions of degree $0$ are just the constants, as usual.)

There is also a noncommutative analogue: noncommutative symmetric functions.

Definitions

Let ${\Lambda }_n$ be the ring consisting of polynomials in $n$ variables $x_1,\dots ,x_n$ that are invariant under all permutations of the variables; these are the symmetric functions in $n$ variables or symmetric polynomials in $n$ variables. The rings ${\Lambda }_n$ are graded by degree in the usual way, and there are homomorphisms of graded rings

$${\Lambda }_{n+1}\hookrightarrow {\Lambda }_n,$$\Lambda_{n+1} \hookrightarrow \Lambda_{n} ,

given by setting the $(n+1)$st variable equal to zero. Taking the limit (in the category-theoretic sense) of these rings ${\Lambda }_n$ as $n\to \mathrm{\infty }$, we get the graded ring ${\Lambda }_{\mathrm{\infty }}$. This is usually called $\Lambda$, 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, $\Lambda$ can be constructed as a colimit of rings using the homomorphisms

$${\Lambda }_n\hookrightarrow {\Lambda }_{n+1},$$\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) $k$, so we may write $\Lambda (k)$ to be precise.

A query about the Hazewinkel’s description of the construction of $\Lambda$ 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, $\Lambda$, has many interesting properties. For example, it is the free $\lambda$-ring on one generator (a coincidence of notation that has not been ignored). It is also a plethory. There are various bases of $\Lambda$ whose elements are in natural one-to-one correspondence with Young diagrams.

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

$$[\mathbb{P},{\mathrm{Vect}}_k]$$[\mathbb{P}, Vect_{k}]

where $\mathbb{P}$ is the groupoid of finite sets and bijections, and ${\mathrm{Vect}}_k$ is the category of vector spaces over a field $k$ of characteristic zero. The category of $k$-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 $\Lambda$.

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

$$F:\mathbb{P}\to {\mathrm{Vect}}_k$$F: \mathbb{P} \to Vect_{k}

with the special property that $F(n)$ is finite-dimensional for all $n$ and $0$ for $n$ 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 $\Lambda$. For more on this, see Schur functor.

References

• Michiel Hazewinkel, Witt vectors, Part 1. arXiv/0804.3888

The classification described above of irreducible $S_n$-modules over $ℂ$ also works unchanged for any algebraically closed field $k$ of characteristic zero. It also works when $k$ has characteristic $p$ and $n$ is not divisible by $p$. 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 $k$ is a field of characteristic $p>0$ then the group algebra of the group $G$ is semisimple iff $p$ doesn't divide the order of $G$.

Apart from this, the field matters a lot. There is a construction that gives all irreducible $k{S}_n$-modules for any field $k$, field, completely analogous to the Specht module construction over $ℂ$. 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:

• Michael 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 $S_n$.