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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 2 / 13
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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 3 / 13
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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 4 / 13
) ∼ = 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 ). S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 5 / 13
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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 6 / 13
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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 7 / 13
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). S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 9 / 13
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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 10 / 13
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). S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 11 / 13
⊗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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 12 / 13
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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 13 / 13
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]. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 14 / 13
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. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 15 / 13
∈ 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). S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 16 / 13