Springer Science+Business Media New York (2016)
COMPARISON OF CANONICAL BASES FOR
SCHUR AND UNIVERSAL ENVELOPING ALGEBRAS
Department of Mathematics
University of Virginia,
Charlottesville, VA, USA
Abstract. We show that the canonical bases in
) and the Schur algebra are
compatible; in fact we extend this result to p-canonical bases. This follows immediately
from a fullness result for a functor categorifying this map. In order to prove this result,
we also explain the connections between categoriﬁcations of the Schur algebra which arise
from parity sheaves on partial ﬂag varieties, singular Soergel bimodules and Khovanov
and Lauda’s “ﬂag category,” which are of some independent interest.
Numerous algebras and representations that appear in Lie theory have bases
which are called “canonical”. There are a variety of arguments for the importance
of these bases, but surely one of their most desirable properties is that these bases
match under natural maps of algebras.
The example we shall consider in this paper is the natural projection φ : U
(d, n) from the quantized universal enveloping algebra to the q-Schur algebra.
In fact, it is more natural to replace U
) with the modiﬁed quantum group
which has the idempotents 1
for the diﬀerent weights λ added. In this case, the
Schur algebra S
(d, n) can be thought of as the quotient of
U which kills 1
cannot be written in the form λ = (a
, . . . , a
= d and a
U is endowed with a canonical basis
B deﬁned by Lusztig [Lus93],
and the Schur algebra S
(d, n) with a canonical basis
(also called IC basis) given
by realizing it as the algebra of GL
)-invariant functions on the space of pairs
of ﬂags of length n (of all possible dimensions) in the space F
, and considering
the functions attached to the IC sheaves smooth along GL
Theorem A. Under the map φ, the set
B \ (
B ∩ ker φ) is sent bijectively to
It’s worth noting that this theorem was proven in [SV00]. However, there it is
submerged as a special case of a more general theorem, and new techniques have
Supported by the NSF under Grant DMS-1151473 and the Alfred P. Sloan Founda-
Received August 13, 2015. Accepted July 6, 2016.
Corresponding Author: B. Webster, e-mail: firstname.lastname@example.org.
Vol. 22, No.
, 2017, pp.
Published online August 1, 2016.