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Archive for the ‘invariant theory’ Category

If V is a finite-dimensional complex vector space, then the symmetric group S_n naturally acts on the tensor power V^{\otimes n} by permuting the factors. This action of S_n commutes with the action of \text{GL}(V), so all permutations \sigma : V^{\otimes n} \to V^{\otimes n} are morphisms of \text{GL}(V)-representations. This defines a morphism \mathbb{C}[S_n] \to \text{End}_{\text{GL}(V)}(V^{\otimes n}), and a natural question to ask is whether this map is surjective.

Part of Schur-Weyl duality asserts that the answer is yes. The double commutant theorem plays an important role in the proof and also highlights an important corollary, namely that V^{\otimes n} admits a canonical decomposition

\displaystyle V^{\otimes n} = \bigoplus_{\lambda} V_{\lambda} \otimes S_{\lambda}

where \lambda runs over partitions, V_{\lambda} are some irreducible representations of \text{GL}(V), and S_{\lambda} are the Specht modules, which describe all irreducible representations of S_n. This gives a fundamental relationship between the representation theories of the general linear and symmetric groups; in particular, the assignment V \mapsto V_{\lambda} can be upgraded to a functor called a Schur functor, generalizing the construction of the exterior and symmetric products.

The proof below is more or less from Etingof’s notes on representation theory (Section 4.18). We will prove four versions of Schur-Weyl duality involving \mathfrak{gl}(V), \text{GL}(V), and (in the special case that V is a complex inner product space) \mathfrak{u}(V), \text{U}(V).

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Let G be a group and let

\displaystyle V = \bigoplus_{n \ge 0} V_n

be a graded representation of G, i.e. a functor from G to the category of graded vector spaces with each piece finite-dimensional. Thus G acts on each graded piece V_i individually, each of which is an ordinary finite-dimensional representation. We want to define a character associated to a graded representation, but if a character is to have any hope of uniquely describing a representation it must contain information about the character on every finite-dimensional piece simultaneously. The natural definition here is the graded trace

\displaystyle \chi_V(g) = \sum_{n \ge 0} \chi_{V_n}(g) t^n.

In particular, the graded trace of the identity is the graded dimension or Hilbert series of V.

Classically a case of particular interest is when V_n = \text{Sym}^n(W^{*}) for some fixed representation W, since V = \text{Sym}(W^{*}) is the symmetric algebra (in particular, commutative ring) of polynomial functions on W invariant under G. In the nicest cases (for example when G is finite), V is finitely generated, hence Noetherian, and \text{Spec } V is a variety which describes the quotient W/G.

In a previous post we discussed instead the case where V_n = (W^{*})^{\otimes n} for some fixed representation W, hence V is the tensor algebra of functions on W. I thought it might be interesting to discuss some generalities about these graded representations, so that’s what we’ll be doing today.

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The theory of symmetric functions, which generalizes some ideas that came up in the previous discussion of Polya theory, can be motivated by thinking about polynomial functions of the roots of a monic polynomial P(x) \in \mathbb{Z}[x]. Problems on high school competitions often ask for the sum of the squares or the cubes of the roots of a given polynomial, and those not familiar with symmetric functions will often be surprised that these quantities end up being integers even though the roots themselves aren’t integers. The key is that the sums being asked for are always symmetric in the roots, i.e. invariant under an arbitrary permutation. The coefficients of a monic polynomial are the elementary symmetric polynomials of its roots, which we are given are integers. It follows that any symmetric polynomial that can be written using integer coefficients in terms of the elementary symmetric polynomials must in fact be an integer, and as we will see, every symmetric polynomial with integer coefficients has this property.

These ideas lead naturally to the use of symmetric polynomials in Galois theory, but combinatorialists are interested in symmetric functions for a very different reason involving the representation theory of S_n. This post is a brief overview of the basic results of symmetric function theory relevant to combinatorialists. I’m mostly following Sagan, who recommends Macdonald as a more comprehensive reference.

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