Seminar on Schur functors and symmetric functions (Winter 2012/13)
Here one can find some information on the seminar.
The seminar meets Wednesday 10:30  12:00 in room V5227.
Organisers are Andrew Hubery and Henning Krause.
The subject
The aim of this seminar is to introduce the theory of Schur functors, with an aim of being able to compute these in specific applications.
Let K be a commutative ring. We recall that the nth tensor power E⊗n of a module E represents the functor sending a module M to the set (in fact Kmodule) of multilinear maps En→M. Similarly we can define the nth exterior power ⋀n(E) as the module representing the functor of alternating multilinear maps, and the nth symmetric power Symn(E) as representing the functor of symmetric multilinear maps. The Schur functors generalise these concepts: for each partition λ one obtains a module Sλ(E) representing a certain class of multilinear maps such that S(1n)(E)≅⋀n(E) and S(n)(E)≅Symn(E). We can then regard the map E↦Sλ(E) as a functor on the category of Kmodules, even on the category of finitelygenerated free Kmodules.
Classically the importance of these functors arose from studying polynomial representations of a general linear group over the complex numbers. In particular, given a complex vector space E of dimension m and a partition λ having at most m rows, then the module Sλ(E) is an irreducible representation of GL(E) having highest weight λ, and all irreducible polynomial representations arise in this way. One can then get any rational representation by tensoring with some power of the determinant representation.
By SchurWeyl duality there is also a close link with the representation theory of the symmetric group. Over the complex numbers, the irreducible representations of the symmetric group Sn are given by the Specht modules Spλ, where λ ranges over the partitions of n. Moreover, one can naturally identify the representation ring Rn of Sn with the ring of symmetric polynomials in n variables, under which the Specht module Spλ is sent to the Schur polynomial sλ. In fact, one can do better: the direct sum R=⨁nRn has the structure of a ring whereby the multiplication is given by induction. In other words, if M is an Smrepresentation and N is an Snrepresentation, then M⊗N is naturally an Sm×Snrepresentation, which we can induce up to get an Sm+nrepresentation. Then there is a ring isomorphism between R and the ring of symmetric functions Λ, which is even an isometry using the natural innerproducts on both rings.
Now, if E is a complex vector space of dimension m and λ is a partition of n having at most m rows, then the irreducible GL(E)representation Sλ(E) is naturally isomorphic to the representation E⊗n⊗SnSpλ. It follows that the character χλ of Sλ(E) satisfies χ(g)=sλ(x1,…,xm), where the xi are the eigenvalues of g on E.
Now let F be another complex vector space, of dimension n, and set A:=Sym∗(E⊗F∨), a polynomial ring in mnvariables, which we can regard as the coordinate algebra of the variety of maps Hom(E,F), or equivalently of all n×m matrices. The rth determinantal ideal Ir⊲A is the ideal generated by all rminors of matrices, so that A/Ir is the coordinate algebra of the affine variety of matrices of rank at most r−1. In this setup, Lascoux constructed a minimal free resolution of the algebra A/Ir, and the ith term of this resolution is a direct sum of modules of the form Sλ(E)⊗Sμ(F∨)⊗A, for certain pairs of partitions (λ,μ).
Shortly afterwards, Kapranov used these techniques to construct a classical tilting bundle on a Grassmannian in characteristic zero, extending the result of Beilinson for projective space. (For a characteristicfree approach using exterior powers see the article by Buchweitz, Leuschke and van den Bergh.) More precisely, let E be an mdimensional complex vector space and let X be the Grassmannian of ldimension subspaces of E. The tautological bundle R on X is the subbundle of the trivial vector bundle X×E given by taking pairs (U,u), where U≤E is an ldimensional subspace and u∈U. Let Q be its dual, yielding the short exact sequence 0→R→X×E→Q→0. Then T:=⨁Sλ(Q) is a classical tilting bundle on X, where the sum is taken over all partitions having at most l rows and at most m−l columns.
Schedule
Multilinear Algebra
 Representable functors. Tensor, exterior and symmetric
products. (17 Oct, Cosima Aquilino)
 Tensor, exterior and symmetric algebras. (24 Oct, Rebecca Reischuk)
 Divided powers. (31 Oct, Henning Krause)
 Schur and Weyl functors. (7 Nov, Valentin Katter)
 Formulae of Cauchy, Pieri and LittlewoodRichardson. (14 Nov, Philipp Lampe)
Characteristic Zero Theory
 Symmetric functions. Representation ring for symmetric
groups. (21 Nov, Andrew Hubery)
 Polynomial functors and plethysm. (28 Nov, Phillip Linke)
 SchurWeyl duality. (19 Dec, Yong Jiang)
Koszul Complexes
 Beilinson's tilting complex for projective space. (16 Jan, Shawn Baland)
References
 K. Akin, D. A. Buchsbaum and J. Weyman, Schur functors and Schur
complexes, Adv. in Math. 44 (1982), no. 3, 207278.
 J. Baez, Schur Functors, The nCategory Café, April 12, 2007.
 A. A. Beĭlinson, Coherent sheaves on P^{n} and problems in linear algebra. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 6869.
 N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3. (French) Hermann, Paris 1970 xiii+635 pp.

N. Bourbaki, Éléments de mathématique. (French) [Elements of mathematics] Algèbre. Chapitres 4 à 7. [Algebra. Chapters 4–7] Lecture Notes in Mathematics, 864. Masson, Paris, 1981. vii+422 pp.
 R.O. Buchweitz, G. J. Leuschke, M. van den Bergh, A characteristic free tilting bundle for Grassmannians, arXiv:1006.1633.

W. Fulton, Young Tableaux, with Applications to Representation
Theory and Geometry, Cambridge University Press, 1997.

W. Fulton and J. Harris, Representation theory. A first course,
SpringerVerlag, 1991.
 M. M. Kapranov, On the derived categories of coherent sheaves on some
homogeneous spaces, Invent. Math. 92 (1988), 479508.
 A. Lascoux, Syzygies des variétés déterminantales. (French)
Adv. in Math. 30 (1978), no. 3, 202237.
 I. G. Macdonald, Symmetric functions and Hall polynomials, second
edition, Oxford Univ. Press, 1995.

J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge University Press, 2003.