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 V5-227.

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 $n$ -th tensor power $E^{\otimes n}$ of a module $E$ represents the functor sending a module $M$ to the set (in fact $K$ -module) of multilinear maps $E^n\to M$ . Similarly we can define the $n$ -th exterior power $\bigwedge^n(E)$ as the module representing the functor of alternating multilinear maps, and the $n$ -th symmetric power $\mathrm{Sym}^n(E)$ as representing the functor of symmetric multilinear maps. The Schur functors generalise these concepts: for each partition $\lambda$ one obtains a module $S^\lambda(E)$ representing a certain class of multilinear maps such that $S^{(1^n)}(E)\cong\bigwedge^n(E)$ and $S^{(n)}(E)\cong\mathrm{Sym}^n(E)$ . We can then regard the map $E\mapsto S^\lambda(E)$ as a functor on the category of $K$ -modules, even on the category of finitely-generated free $K$ -modules.

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 $\lambda$ having at most $m$ rows, then the module $S^\lambda(E)$ is an irreducible representation of $\mathrm{GL}(E)$ having highest weight $\lambda$ , 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 Schur-Weyl 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 $S_n$ are given by the Specht modules $\mathrm{Sp}^\lambda$ , where $\lambda$ ranges over the partitions of $n$ . Moreover, one can naturally identify the representation ring $R_n$ of $S_n$ with the ring of symmetric polynomials in $n$ variables, under which the Specht module $\mathrm{Sp}^\lambda$ is sent to the Schur polynomial $s_\lambda$ . In fact, one can do better: the direct sum $R=\bigoplus_nR_n$ has the structure of a ring whereby the multiplication is given by induction. In other words, if $M$ is an $S_m$ -representation and $N$ is an $S_n$ -representation, then $M\otimes N$ is naturally an $S_m\times S_n$ -representation, which we can induce up to get an $S_{m+n}$ -representation. Then there is a ring isomorphism between $R$ and the ring of symmetric functions $\Lambda$ , which is even an isometry using the natural inner-products on both rings.

Now, if $E$ is a complex vector space of dimension $m$ and $\lambda$ is a partition of $n$ having at most $m$ rows, then the irreducible $\mathrm{GL}(E)$ -representation $S^\lambda(E)$ is naturally isomorphic to the representation $E^{\otimes n}\otimes_{S_n}\mathrm{Sp}^\lambda$ . It follows that the character $\chi_\lambda$ of $S^\lambda(E)$ satisfies $\chi(g)=s_\lambda(x_1,\ldots,x_m)$ , where the $x_i$ are the eigenvalues of $g$ on $E$ .

Now let $F$ be another complex vector space, of dimension $n$ , and set $A:=\mathrm{Sym}^*(E\otimes F^\vee)$ , a polynomial ring in $mn$ -variables, which we can regard as the co-ordinate algebra of the variety of maps $\mathrm{Hom}(E,F)$ , or equivalently of all $n\times m$ matrices. The $r$ -th determinantal ideal $I_r\lhd A$ is the ideal generated by all $r$ -minors of matrices, so that $A/I_r$ is the co-ordinate algebra of the affine variety of matrices of rank at most $r-1$ . In this set-up, Lascoux constructed a minimal free resolution of the algebra $A/I_r$ , and the $i$ -th term of this resolution is a direct sum of modules of the form $S^\lambda(E)\otimes S^\mu(F^\vee)\otimes A$ , for certain pairs of partitions $(\lambda,\mu)$ .

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 characteristic-free approach using exterior powers see the article by Buchweitz, Leuschke and van den Bergh.) More precisely, let $E$ be an $m$ -dimensional complex vector space and let $X$ be the Grassmannian of $l$ -dimension subspaces of $E$ . The tautological bundle $\mathcal R$ on $X$ is the subbundle of the trivial vector bundle $X\times E$ given by taking pairs $(U,u)$ , where $U\leq E$ is an $l$ -dimensional subspace and $u\in U$ . Let $\mathcal Q$ be its dual, yielding the short exact sequence $0\to\mathcal R\to\mathcal X\times E\to\mathcal Q\to 0$ . Then $\mathcal T:=\bigoplus S^\lambda(\mathcal Q)$ 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 Littlewood-Richardson. (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)
Schur-Weyl 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, 207--278.
J. Baez, Schur Functors, The n-Category Café, April 12, 2007.
A. A. Beĭlinson, Coherent sheaves on Pⁿ and problems in linear algebra. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68--69.
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, Springer-Verlag, 1991.
M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479--508.
A. Lascoux, Syzygies des variétés déterminantales. (French) Adv. in Math. 30 (1978), no. 3, 202--237.
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.