Boulder-2013-April-13

Rational Schur Algebras
S.R. Doty
Loyola University Chicago
April 13, 2013
S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 1 / 13

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Classical Schur–Weyl duality
Let k be an inﬁnite ﬁeld. Let V be a vector space over k with
dim
k
V = n, and V ∗ = Hom
k
(V , k) its linear dual.
Given a positive integer r, form tensor space V ⊗r . Consider the
commuting actions of the groups GL(V ), acting diagonally on the left, and
the symmetric group Sr , acting on the right by place-permutations. We
have representations
kGL(V ) Φ
−→ End
k
(V ⊗r ) Ψ
←− kSr
corresponding to our actions. Since the actions commute, we have
inclusions
Φ(kGL(V )) ⊆ EndSr
(V ⊗r ); Ψ(kSr ) ⊆ EndGL(V )
(V ⊗r )
of the envelope (image) of each action in the centralizer algebra of the
other action.

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Classical Schur–Weyl duality
The following result was proved by Issai Schur over the complex field C,
and later generalized.
Theorem (Schur 1927, DeConcini–Procesi 1976, J.A. Green 1980)
The inclusions on the previous slide are actually equalities. That is,
Φ(kGL(V )) = EndSr
(V ⊗r ); Ψ(kSr ) = EndGL(V )
(V ⊗r )
for any infinite field k.
Remark: The n ≥ r case of this theorem was proved by Carter–Lusztig in
1974.

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Classical Schur algebras
Definition (J.A. Green, SLN 830)
The Schur algebra is the centralizer algebra S
k
(n, r) = EndSr
(V ⊗r ). This
makes sense for any commutative ring k.
When k is an infinite field (or even a “big enough” finite one) the algebra
S
k
(n, r) provides a bridge connecting the modular representation theories
of general linear and symmetric groups.
G.D. James (Math. Z., 1980) proved that the modular decomposition
matrices of Sn are submatrices of those for GLn
∼
= GL(V ).
K. Erdmann (J of Algebra, 1996) proved that if we knew the modular
decomposition matrices of Sr for all r then we could recover them for
any given GLn.
Friedlander and Suslin (Inventiones, 1997) used Schur algebras to
prove finite generation of cohomology of finite group schemes.

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Reformulation of Schur–Weyl duality
Let G
C
= GL(V
C
) ∼
= GLn(C) be the general linear Lie group.
Then g = gln
(C) is its Lie algebra. Let U = U(g) be the universal
enveloping algebra, and U
Z
(g) its Kostant integral form (generated by
divided powers of root vectors in a Chevalley basis).
Put U
k
(g) = k ⊗
Z
U
Z
. This is the so-called “hyperalgebra” or algebra of
distributions on the algebraic group (scheme) GL(V ) ∼
= GLn(k) over k.
We have an action of U
k
(g) on V ⊗r which commutes with the place
permutation action of Sr , so we have representations
U
k
(g) Φ
−→ End
k
(V ⊗r ) Ψ
←− kSr .
Theorem (Carter–Lusztig, 1974 (most cases))
For any commutative ring k, Φ(U
k
(g)) = EndSr
(V ⊗r ) and
Ψ(kSr ) = EndUk
(g)
(V ⊗r ).

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The Brauer algebra
A Brauer r-diagram is an undirected graph with 2r vertices and r edges,
such that each vertex is the endpoint of precisely one edge.
By convention, such a graph is usually drawn in a rectangle with r vertices
each equally spaced along the top and bottom edges of the rectangle. For
example, the picture below
depicts a Brauer 8-diagram.
Let k be a commutative ring. Let Br (δ) be the vector space over k with
basis the r-diagrams, where δ ∈ k.

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The Brauer algebra (multiplication rule)
To describe the multiplication rule in Br (δ), it is convenient to introduce
the notations Top(d) and Bot(d) for the sets of vertices along the top and
bottom edges of a diagram d.
Then multiplication works as follows. Given r-diagrams d1 and d2, place
d1 above d2 and identify the vertices in Bot(d1) in order with those in
Top(d2).
The resulting graph consists of r paths whose endpoints are in
Top(d1) ∪ Bot(d2), along with a certain number, say M, of cycles which
involve only vertices in the middle row. Let d be the r-diagram whose
edges are obtained from the paths in this graph and whose vertices are
Top(d1) ∪ Bot(d2).
Then the product of d1 and d2 in Br (δ) is given by d1d2 = δMd.

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Example of multiplication rule
Multiplying the diagrams below (in the indicated order)
produces the following result in Br (δ):
δ ·

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Action of Br
(n) on V ⊗r
Br (δ) is generated by the diagram e := e1,2 given by
e = · · ·
along with the permutation diagrams (with no through strings) in the
symmetric group algebra kSr , which act on V ⊗r by place-permutation.
Specialize δ to n = dim
k
V . To specify an (right) action of Br (n) on V ⊗r
it is only necessary to say how the generator e acts. Deﬁne:
(vi1
⊗ vi2
⊗ · · · ⊗ vir
) · e = δi1,i
2
n
j=1
vj ⊗ vj ⊗ vi3
⊗ · · · ⊗ vir
where {v1, . . . , vn} is the canonical basis for V ∼
= kn, and where
j := n + 1 − j. The δ is Kronecker’s delta function (not the parameter).

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Schur–Weyl duality (orthogonal case)
Let k be an infinite field of characteristic = 2. Let O(V ) be the
orthogonal group of linear operators on V preserving the bilinear form
defined by vi , vj = δi,j , where j = n + 1 − j as in previous slide.
Remark: In Lie theory, it is often more convenient to use such a defining
form when dealing with orthogonal groups or their Lie algebras. For
instance, with this choice of defining form, the maximal torus in O(V ) is
just the restriction of the diagonal torus in GL(V ).
Theorem (Brauer 1927 (for k = C); D–Hu PLMS 2009 (in general))
Let k be an infinite field of characteristic = 2. The natural action of O(V )
on V ⊗r commutes with the action of Br (n). The envelope of each action
is equal to the centralizer of the other.

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Walled Brauer algebra
Definition
A through string is an edge in a Brauer diagram d with one endpoint in
Top(d) and one in Bot(d). All other edges are non-through or horizontal.
Let r, s be given non-negative integers. An (r, s)-diagram is an m-diagram
(m = r + s) in which we imagine an infinite vertical line (the wall)
separating the first r vertices in each row from the last s, such that:
through strings never intersect the wall;
non-through strings always do.
Let Br,s(δ) be the subalgebra of Bm(δ), where m = r + s, spanned by the
set of (r, s)-diagrams. This is the walled Brauer algebra (Benkart et al,
J. Algebra, 1994).

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Schur–Weyl duality (mixed tensor case)
Let V r,s := V ⊗r ⊗ (V ∗)⊗s. Call this mixed tensor space. Let Br,s(n) act
(on the right) on V r,s by identifying V and V ∗ via the defining bilinear
form , . In other words, it acts by restriction as a subalgebra of Bm(n)
on V ⊗m where m = r + s.
This action commutes with the natural (left) action of GL(V ) on V r,s. So
we have the following representations
kGL(V ) Φ
−→ End
k
(V r,s) Ψ
←− Br,s(n).
Theorem (Benkart et al (k = C); Dipper–D–Stoll (general); Tange
(general))
Let k be an infinite field. Then the envelope of each action equals the
centralizer algebra for the other.
Remark: If we replace GL(V ) by its hyperalgebra U
k
(gln
) then the above
is true for any commutative ring k.

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The rational Schur algebra
Definition (Dipper–D, Rep. Theory 2008)
The rational Schur algebra over k is S
k
(n; r, s) := EndBr,s (n)
(V r,s). This
makes sense for any commutative ring k.
The category of finite-dimensional modules for S
k
(n; r, s) is the category
of rational representations of GL(V ) of bidegee (r, s), in a suitable sense.
(Related to “rational tableaux” introduced by Stembridge 1987; see also
R.C. King 1970.)
Theorem (Dipper–D, Donkin 2012)
If k is a field then S
k
(n; r, s) is quasihereditary over k.
Remarks: (1) Donkin (recent preprint) closes a gap in the proof given in
the Dipper–D paper.
(2) If s = 0 then S
k
(n; r, s) = S
k
(n, r) is the classical Schur algebra.

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Quantized walled Brauer algebra
Two preprints [DDS1, DDS2] on arXiv by Dipper–D–Stoll (first versions
go back to 2008).
The first paper [DDS1] investigates an algebra Bn
r,s
(a, λ, δ) over an
arbitrary commutative ring k. This algebra, a “deformation” of the walled
Brauer algebra, is defined by generators and relations.
Bn
r,s
(a, λ, δ) was first studied in Rob Leduc’s dissertation (Madison 1994)
over k = C; see also Kosuda–Murakami (Osaka Math J. 1993).
[DDS1] gives a description of Bn
r,s
(a, λ, δ) in terms of oriented tangles in
the sense of Kauffman (TAMS 1990), and finds a basis consisting of
oriented tangles, which is in bijection with the basis of (r, s)-diagrams of
the walled Brauer algebra Br,s(δ). The paper finishes by proving one-half
of the following result, the other half of which is proved in [DDS2].

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Schur–Weyl duality for quantized walled Brauer algebra
For any invertible q ∈ k, write Bn
r,s
(q) := Bn
r,s
(q−1 − q, qn, [n]q), where
q ∈ k. This specialization is necessary in order to get an action on mixed
tensor space V r,s. Let Uq(gln
) be the quantized enveloping algebra (over
k at parameter q) of gln
.
Theorem (DDS1, DDS2)
Let k be an arbitrary commutative ring. Then there are commuting
actions of Bn
r,s
(q) and Uq(gln
) on mixed tensor space V r,s, and the
envelope of each action equals the centralizer for the other.
This includes the case where q is a root of unity.
Various earlier results are special cases of this more general result. The
proof uses the quantized version of classical Schur–Weyl duality
established by Jimbo 1986; Du–Parshall–Scott 1998.

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The quantized rational Schur algebra
Definition
For any invertible q ∈ k, where k is a commutative ring, put
Sq(n; r, s) := EndBn
r,s
(q)
(V r,s). This is the rational q-Schur algebra.
If we take q = 1 then we recover the ordinary rational Schur algebra
S
k
(n; r, s) defined earlier in this lecture.
When s = 0, the algebra Sq(n; r, 0) is the original q-Schur algebra Sq(n, r)
defined by Dipper–James (PLMS 1989).
The paper [DDS2] obtains a “bideterminant” basis of the algebra
Sq(n; r, s). When k is a field, the algebra is quasihereditary. The finite
dimensional modules for Sq(n; r, s) are the type 1 representations of
Uq(gln
) in bidegree (r, s).

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Boulder-2013-April-13

Boulder-2013-April-13

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