Geometry of Schubert Varieties RepNet Workshop Chris Spencer Ulrich Thiel University of Edinburgh University of Kaiserslautern 24 May 2010
Flag Varieties Throughout, let k be a fixed algebraically closed field and V a finite-dimensional k-vector space. A flag in V is a nested sequence of strictly increasing subvector spaces: F := (0 = F 0 < F 1 < < F r = V ) If r = dim V we speak of a complete flag, otherwise we call F a partial flag. For d N r we set the set of d-flags. Fl(V, d) := {F dim(f i /F i 1 ) = d i, i = 1... r}
Flag Varieties When is Fl(V, d) a variety? Some examples: d = (1, dim V 1) we obtain Fl(V, d) = P(V ) d = (l, dim V l) we have Fl(V, d) = Gr(V, l), the Grassmannian of l-dimensional subspaces of V Promising! We could proceed by embedding Fl(V, d) in a product of Grassmannians Fl(V, d) Gr(V, dim F 1 ) Gr(V, dim F 2 )... Gr(V, dim F r ). (F 0 <F 1 <... < F r ) (F 1, F 2,..., F r ) ut let us give a description that is easier to generalize (of course the same up to isomorphism):
Flag Varieties We clearly have a natural action of GL(V ) on Fl(V, d) by g F = (0 = g(f 0 ) < g(f 1 ) < < g(f r 1 ) < g(f r ) = V ) and this action is transitive. So for any fixed flag F Fl(V, d) we have Fl(V, d) = GL(V )/Stab(F ) and we define this to be an isomorphism of varieties (well-defined because two stabilizers are conjugate).
Flag Varieties Example For V = k n and the standard flag we have E = (0 < e 1 < e 1, e 2 <... < e 1, e 2,..., e n = k n ) Stab(E ) = := {upper triangular matrices} := standard orel subgroup and so Fl(k n, (1, 1,..., 1)) = GL n (k)/
Coxeter groups Let (W, S) be a Coxeter system with W finite, i.e. W is a finite group with presentation S (s i s j ) m ij = 1, where m ij N { } with m ii = 1 and m ij = m ji 2 for i j. Here, (s i s j ) = 1 denotes the empty relation. Definition l : W N, l(w) := min #{l w = s 1 s l with s i S} is the length function. Any expression w = s 1 s l(w) with s i S is called a reduced expression of w. R := {wsw 1 w W, s S} are the reflections of (W, S). If u 1 w R and l(u) < l(w), one writes u w and says u covers w. (Note: u 1 w R if and only if wu 1 R) If there exists a sequence u = u 0 u 1 u r = w, one writes u w.
Coxeter groups Theorem (i) is a partial order on W (ruhat order). (ii) If w = s 1 s l is a reduced expression of w, then u w if and only if a subexpression of s 1 s l is equal to u. (iii) If u w, then there exists a sequence u = u 0 u 1 u r = w with l(u i+1 ) = l(u i ) + 1. (iv) 1 w for all w W. (v) is directed, i.e. if u, v W, then u w, v w for some w W. (vi) There exists a unique element w 0 W of maximal length. Moreover, w w 0 for all w W.
Coxeter groups Our primary example of a Coxeter system: Theorem (S n, S) with S := {s i 1 i n 1} and s i := (i, i + 1) is a Coxeter system. Some properties: (i) l(w) = #{(i, j) i < j, w(i) > w(j)} for all w S n. (ii) w 0 = (n n 1 1).
Coxeter groups Definition The ruhat graph of (W, S) is the directed graph with vertices W and arrows given by the covering relation. The ruhat graph of S 3 is: Length 3 2 231 321 312 1 132 213 0 123
Schubert varieties in flag varieties Throughout, we fix G := GL n (k). We denote by the group of upper triangular matrices in G and by T the group of diagonal matrices in G. Definition (i) N G (T ) is the group of monomial matrices. Hence, W := N G (T )/T = S n canonically. The permutation matrices in G form a complete set of representatives {ẇ} of W. (ii) S := {s i 1 i < n}, where s i := (i, i + 1). (iii) Φ := {χ ij 1 i, j n, i j}, where χ ij : T k, diag(t 1,..., t n ) t i t 1 j. (iv) Φ + := {χ ij 1 i < j n}. (v) s χij := s ij := (i, j). (vi) W acts on Hom(T, k ) by (w.χ)(t) := χ(t w ).
Schubert varieties in flag varieties Definition (i) U := the group of upper unitriangular matrices. (ii) := the group of lower triangular matrices. (iii) U := the group of lower unitriangular matrices. (iv) U w := U wu w 1 for w W. (v) Φ w := {α Φ+ w 1 α Φ + } = {χ ij i < j, w 1 (i) > w 1 (j)}. (vi) u χij := u ij : k G, c 1 + ce ij.
Schubert varieties in flag varieties Theorem (i) G = w W w (ruhat decomposition). (ii) The map U w w given by (u, b) uẇb is an isomorphism. (iii) G/ = w W w/. We call C w := w/ a Schubert cell in G/. (iv) The map U w C w given by u uẇ is an isomorphism. (v) Let α 1,..., α l be the roots of Φ w (any ordering). Then the map A l(w) U w, (a 1,..., a l ) u α1 (a 1 ) u αl (a l ), is an isomorphism of varieties. Hence, A l(w) = C w.
Schubert varieties in flag varieties Definition For each w W we set X w := w/ G/ and refer to this as the Schubert variety of w. Theorem The following holds: (i) X w is a projective variety and dim X w = l(w). (ii) X w = u w C u, where denotes the ruhat order. (iii) X w = u w X u. (iv) X u X w if and only if u w. (v) X w0 = G/ and X id = pt.
Let n = 4. For w = (2413) we have 1 0 1 0 U w = 0 1 0 0 0 0 1 =A 3 =A l(w) C w = 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0. We have Φ w = {χ 12, χ 14, χ 34 } and indeed the map A 3 U w defined above as is an isomorphism. (a 1, a 2, a 3 ) u 12 (a 1 )u 14 (a 2 )u 34 (a 3 ) = 1 + a 3 E 34 + a 2 E 14 + a 1 E 12 1 a 1 0 a 2 = 0 1 0 0 0 0 1 a 3 0 0 0 1
Schubert varieties in flag varieties Let us consider an example now: 6 9 2 1 6 3 2 7 1 1 0 1 0 6 2 1 1 I+II 1 0 0 0 1 I+IV 0 6 2 1 0 3 1 0 0 3 1 0 6 1 1 7 1 II+III 1 0 0 0 0 2 0 1 7 III+IV 0 1 0 0 1 3 II 6 1 1 0 1 0 0 0 0 2 0 1 0 1 0 0 So this is in C 2413 with coordinates (6, 1, 2). 6 1 2 7 1 0 0 0 0 2 2 1 0 1 1 0
Schubert varieties in flag varieties We can also determine the decomposition M = uẇb with u U w and b : The column operations give b 1, ẇ can be read off from M and u computed from M, b, ẇ: 1 1 0 1 0 0 1 0 1 6 0 1 0 3 1 0 1 0 0 0 0 1 0 0 b = 0 0 1 7 0 0 0 1, ẇ = 0 0 0 1 0 1 0 0 u = 0 0 1 2 0 0 1 0
The moment graph of G/ The maximal torus T acts canonically on G/. The root morphisms u α and their property tu α (c)t 1 = u α (α(t)c) are the central ingredients in the proof of the following theorem: Theorem (i) The set of T -fixed points is {w w W }, i.e. (G/) T =W. (ii) The one-dimensional T -orbits are precisely the O w,α := {u α (x)w x k } = Tu α (1)w for w W and α Φ w. (iii) O w,α C w for all α Φ w. (iv) If l(w) = 1, then O w,α = X w for the unique α Φ w. In particular, all Schubert curves in G/ are among the closures of the one-dimensional T -orbits.
The moment graph of G/ There is a T -equivariant embedding G/ P(V ) for some V, where the T -action on P(V ) is induced by a linear action of T on V. Hence, if O is a one-dimensional T -orbit, then there exists a T -equivariant isomorphism ϕ : O P 1, where O is the closure of O and the T -action on P 1 is induced by a linear action on k 2. In particular, O is obtained from O by adding the two T -fixed points ϕ 1 (0) and ϕ 1 ( ). Theorem The two T -fixed points of the closure of O w,α are w and s α w.
The moment graph of G/ Combining the results above with the following lemma: Lemma Let (W, S) be a Coxeter system with W finite. Let w W and α Φ +. Then l(ws α ) < l(w) if and only if w(α) Φ +. we arrive at the following amazing result: Theorem The moment graph of G/ with respect to the action of T is precisely the undirected ruhat graph of (W, S).
The moment graph of G/ Let n = 3. Then Φ + = {χ 12, χ 13, χ 23 }, Φ 123 =, Φ 132 = {χ 23}, Φ 213 = {χ 12}, Φ 231 = {χ 12, χ 13 }, Φ 312 = {χ 13, χ 23 }, Φ 321 = {χ 12, χ 13, χ 23 }. We have s 23 (132) = (123), s 12 (213) = (123), s 12 (231) = (132), s 13 (231) = (213), s 13 (312) = (132), s 23 (312) = (213), s 12 (321) = (312), s 13 (321) = (123), s 23 (321) = (231). Hence, the moment graph of G/ is 321 231 312 132 213 123 which is just the (undirected) ruhat graph of S 3.
ott-samelson resolution X w need not be smooth, although C w always is. Theorem For G = GL n (k) and y S n the dimension of the tangent space T y X w is dim T y X w = #{(i, j) S n y(i, j) w} So for GL 4 (k) we see that X (1,4) is singular: It has dimension l((1, 4)) = 5 but T X (1,4) has dimension 6 corresponding to the permutations (1, 2), (2, 3), (3, 4), (1, 4), (2, 4), (1, 3).
ott-samelson resolution It is an important problem in algebraic geometry to find resolutions of singular spaces. For Schubert varieties this can be done via the ott-samelson varieties. Definition A parabolic subgroup P of G is any closed subgroup containing. If X is a space with a left action, the induced P-space is the quotient P X := (P X )/ := P X /((g, x) (gb 1, bx) b ). P X has a canonical -action via left multiplication and thus for any P 1, P 2,..., P l of parabolics this construction can be iterated: P 1 P2... Pl X := P1 (P2 (... (Pl X )...))
ott-samelson resolution Definition P s is the subgroup of G generated by and s S. This will be referred to as a minimal parabolic. We have P s = s. Consider w W with reduced expression w = s α1 s α2... s αl. We then define the ott-samelson variety Z w as Z w := P sα1 Psα2... (Psαl /). Theorem Z w is a smooth projective variety of dimension l(w). The proof uses the inductive construction of the Z w as Z w = P s Zw for some w < w as well as the following lemma:
ott-samelson resolution Lemma (i) The induced P-space gives a fiber bundle over P/ with fiber X and total space P X : X P X P/ (ii) For any minimal parabolic P s the space P s / is isomorphic to P 1 as a variety. The last line gives us the first step in the induction. Note that P si / corresponds to the flags (0 = E 0 < E 1 <... E i 1 < L < E i+1 <... E n = k n ) and L is uniquely determined by the choice of a line in E i+1 /E i 1 giving the isomorphism with P 1 = P(E i+1 /E i 1 ).
ott-samelson resolution As the name suggests, the Z w will give us the ott-samelson resolution. We can organize the key results into one theorem: Theorem Consider a Schubert variety X w G/. Then the following holds: (i) The morphism Z w X w (p 1,..., p l(w) ) p 1 p 2... p l(w) is a resolution of singularities. (ii) X w is normal. (iii) X w has at most rational singularities. Corollary The singular locus of a Schubert variety has codimension 2.