In mathematics, Schur algebras, named after
Issai Schur, are certain finite-dimensional
algebras closely associated with
Schur–Weyl duality between
general linear and
symmetric groups. They are used to relate the
representation theories of those two
groups. Their use was promoted by the influential monograph of
J. A. Green first published in 1980.[1] The name "Schur algebra" is due to Green. In the modular case (over infinite
fields of positive characteristic) Schur algebras were used by Gordon James and
Karin Erdmann to show that the (still open) problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent.[2] Schur algebras were used by Friedlander and
Suslin to prove finite generation of
cohomology of finite
group schemes.[3]
Construction
The Schur algebra can be defined for any
commutative ring and integers . Consider the
algebra of
polynomials (with coefficients in ) in commuting variables , 1 ≤ i, j ≤ . Denote by the homogeneous polynomials of degree . Elements of are k-linear combinations of monomials formed by multiplying together of the generators (allowing repetition). Thus
Now, has a natural
coalgebra structure with comultiplication and counit the algebra homomorphisms given on generators by
Since comultiplication is an algebra homomorphism, is a
bialgebra. One easily
checks that is a subcoalgebra of the bialgebra , for every r ≥ 0.
Definition. The Schur algebra (in degree ) is the algebra . That is, is the linear dual of .
It is a general fact that the linear
dual of a coalgebra is an algebra in a natural way, where the multiplication in the algebra is induced by dualizing the comultiplication in the coalgebra. To see this, let
and, given linear functionals , on , define their product to be the linear functional given by
The identity element for this multiplication of functionals is the counit in .
Main properties
One of the most basic properties expresses as a centralizer algebra. Let be the space of rank column vectors over , and form the
tensor power
Then the
symmetric group on letters acts naturally on the tensor space by place permutation, and one has an isomorphism
In other words, may be viewed as the algebra of
endomorphisms of tensor space commuting with the action of the
symmetric group.
Various bases of are known, many of which are indexed by pairs of semistandard
Young tableaux of shape , as varies over the set of
partitions of into no more than parts.
In case k is an infinite field, may also be identified with the enveloping algebra (in the sense of H. Weyl) for the action of the
general linear group acting on (via the diagonal action on tensors, induced from the natural action of on given by matrix multiplication).
Schur algebras are "defined over the integers". This means that they satisfy the following change of scalars property:
for any commutative ring .
Schur algebras provide natural examples of quasihereditary algebras[4] (as defined by Cline, Parshall, and Scott), and thus have nice
homological properties. In particular, Schur algebras have finite
global dimension.
Generalizations
Generalized Schur algebras (associated to any reductive
algebraic group) were introduced by Donkin in the 1980s.[5] These are also quasihereditary.
Around the same time, Dipper and James[6] introduced the quantized Schur algebras (or q-Schur algebras for short), which are a type of q-deformation of the classical Schur algebras described above, in which the symmetric group is replaced by the corresponding
Hecke algebra and the general linear group by an appropriate
quantum group.
There are also generalized q-Schur algebras, which are obtained by generalizing the work of Dipper and James in the same way that Donkin generalized the classical Schur algebras.[7]
There are further generalizations, such as the affine q-Schur algebras[8] related to affine
Kac–MoodyLie algebras and other generalizations, such as the cyclotomic q-Schur algebras[9] related to Ariki-Koike algebras (which are q-deformations of certain
complex reflection groups).
The study of these various classes of generalizations forms an active area of contemporary research.
^Karin Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules. Journal of Algebra 180 (1996), 316–320.
doi:
10.1006/jabr.1996.0067MR1375581
^Edward Cline, Brian Parshall, and Leonard Scott, Finite-dimensional algebras and highest weight categories. Journal für die Reine und Angewandte Mathematik [Crelle's Journal] 391 (1988), 85–99.
MR0961165
^Richard Dipper and Gordon James, The q-Schur algebra. Proceedings of the London Math. Society (3) 59 (1989), 23–50.
doi:
10.1112/plms/s3-59.1.23MR0997250
Hermann Weyl, The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939.
MR0000255,
ISBN0-691-05756-7