INTRODUCTION TO SCHUBERT VARIETIES LMS SUMMER SCHOOL, GLASGOW 2018
MARTINA LANINI
1. The birth: Schubert’s enumerative geometry Schubert varieties owe their name to Hermann Schubert, who in 1879 pub- lished his book on enumerative geometry [Schu]. At that time, several people, such as Grassmann, Giambelli, Pieri, Severi, and of course Schubert, were inter- ested in this sort of questions. For instance, in [Schu, Example 1 of §4] Schubert asks 3 1 How many lines in PC do intersect 4 given lines? Schubert’s answer is two, provided that the lines are in general position. How did he get it? And what does general position mean? Assume that we can divide the four lines `1, `2, `3, `4 into two pairs, say {`1, `2} and {`3, `4} such that the lines in each pair intersect in exactly one point and do not touch the other two. Then each pair generates a plane, and we assume that these two planes, say π1 and π2, are not parallel (i.e. the lines are in general position). Then, the planes intersect in a line. If this happens, there are exactly two lines intersecting `1, `2, `3, `4: the one lying in the intersection of the two planes and the one passing through the two intersection points `1 ∩`2 and `3 ∩ `4. Schubert was using the principle of special position, or conservation of num- ber. This principle says that the number of solutions to an intersection problem is the same for any configuration which admits a finite number of solutions (if this is the case, the objects forming the configuration are said to be in general position). Unluckily, the notion of general position is ambigous, and caused several mistakes in solution counting. Making enumerative geometry rigourous was the 15th of Hilbert’s problems and at the origin of the Schubert calculus. We will not be able to discuss this into details
1.1. Schubert’s question translated into modern language. Recall that the complex projective space n is defined as the quotient of n \{(0, 0,..., 0)t} PC C by the equivalence relation ∼ given by x1 x1 x1 . . . n t . ∼ λ . for any λ ∈ C \{0}, . ∈ C \{(0, 0,..., 0) }. xn xn xn 1 2 MARTINA LANINI
t n t We denote the equivalence class of a point (x1, . . . , xn) ∈ C \{(0, 0,..., 0) } by x1 . . . xn The reason why we are using column vectors instead of row vectors will become clear later in Exercise 3.5 Notice that, by definition, a point in n is a line through the origin in n PC C (hence, it is nothing but a 1-dimensional subspace). Let’s get back to Schubert’s question. He was considering lines in 3 , which PC 4 are nothing but 2-dimensional subspaces of C (see §1.1.1). n Definition 1.1. The set of k-dimensional subspaces of C is denoted by Gr(k, n). We will refer to it as the Grassmannian or Grassmann variety. As the name suggests, Gr(k, n) is more than a set: it is a projective algebraic variety. We will get back to this in the next section. Exercise 1.1. Show that for any m there exist k, n such that Gr(k, n) = m. PC
The four given lines are hence four points W1,W2,W3,W4 ∈ Gr(2, 4) and Schubert’s example amounts in determining
#{V ∈ Gr(2, 4) | V ∩ Wi 6= 0 for i = 1, 2, 3, 4}. For i = 1, 2, 3, 4, define
(1) Yi := {V ∈ Gr(2, 4) | V ∩ Wi 6= 0}, T4 so that we are looking for # i=1 Yi. Each one of the Yi’s is a Schubert variety: a variety defined by incidence relations (see Example 2.6). 1.1.1. Little digression on projective geometry. While discussing Schubert’s enu- merative geometric question, we looked at lines in 3 , and we then said that PC 4 they are nothing but (images) of 2-dimensional subspaces in C . In general, a n n+1 k-space in P is the image of a k + 1-dimensional subspace of C . A hyperplane (or an n − 1-space) in n is the zero set of a homogeneous PC polynomial of degree 1:
a0x0 + a1x1 + . . . anxn = 0, n where (a1, . . . , an) ∈ C \{(0,..., 0)}. A k-space in n is the intersection of n − k hyperplanes: PC a x + a x + ... + a x = 0 1,0 0 1,1 1 1,n n a2,0x0 + a2,1x1 + ... + a2,nxn = 0 (2) . . . . . . . . . . an−k,0x0 + an−k,1x1 + ... + an−k,nxn = 0 such that the rank of the matrix (ai,j) is maximal (i.e. rk(ai,j) = n − k). At this point the correspondence n n+1 {k-spaces in PC} ↔ {(k + 1) − dimensional subspaces in C } 3 should be clear: we know that a homogeneous system of equations in m- variables over C, with associate coefficient matrix A, admits a solution space of dimension m − rk(A), so that if we now look for the solution space of the n+1 system (2) in C , we see that it has dimension n + 1 − (n − k) = k + 1.
2. The case of the Grassmann variety 2.1. Basics on varieties. Definition 2.1. An affine (algebraic) variety is the set of solutions to a set of n polynomials in F for some field F. Example 2.1. For example, the graph of y = x2 − x is an affine variety in 2 R , since it is the set of all points (x, y) annihilating the polynomial f(x, y) = y − x2 + x. Definition 2.2. A projective (algebraic) variety is the common zero locus in n of a finite set of homogeneous polynomials in n. PF PF
Observe that if f(x0, . . . , xn) is homogeneous, then deg(f) f(λx0, . . . , λxn) = λ f(x0, . . . , xn), for any λ ∈ F \{0}, so that f(x0, . . . , xn) = 0 if and only if f(λx0, . . . , λxn) = 0 for any λ ∈ F \{0}. Hence it makes sense to talk about zeroes of a homogeneous polynomial in n. PF We will focus on the case F = C, the field of complex numbers, and explain why Gr(k, n) is a complex projective variety.
2.2. A canonical representative in Matn×k. We denote by Matn×k the set of n × k matrices with coefficients in C. n Let (e1, . . . , en) be the standard basis of C (all entries of ei are zero, but the i-th which is a 1). Given a subspace U ∈ Gr(k, n), we want to associate with it a matrix MU ∈ Matn×k in a canonical way. First of all, we can choose a basis for U, say (b1, . . . bk), and can write all bi’s in terms of the standard basis. In this way we obtain k column vectors, that we arrange into an n × k matrix (having as columns the bi’s ). Notice that we can perform column operations to get another matrix, corresponding to another basis of the same subspace, and hence the n same point in Gr(k, n). It follows that any k-dimensional subspace U of C is 2 represented by a matrix MU in column echelon form . If, moreover, we assume that the pivots are ordered left to right, top to bottom, then such a matrix is unique and we denote it by MU .
Example 2.2. Let us assume k = 2, n = 4 and denote by (e1, e2, e3, e4) the 4 standard basis of C . If we assume
(3) U = spanC{8e1 + 4e2 + 2e3, 5e1 + 2e2 + e3},
2A matrix is in column echelon form if any column has a pivotal 1, i.e., a 1 such that it is the first non-zero entry looking from the bottom of such a column and the first non-zero entry looking from the right of the column to whom it belongs. 4 MARTINA LANINI then, 8 5 −2 5 1 5 10 4 2 0 2 0 2 0 2 → → → = MU 2 1 0 1 0 1 01 0 0 0 0 0 0 0 0 2.3. Schubert cells.
Definition 2.3. Let U ∈ Gr(k, n), and let MU ∈ Matn×k be the representing matrix. The index set of the rows where the pivotal ones are is called the position of U, and denoted by pos(U).
Notice that by definition of MU , the position is a strictly increasing sequence of k numbers choosen from the set {1, . . . , n}. Example 2.3. In Example 2.2, we have pos(U) = {1, 3}. k We notice right away that the number of possible positions is n , that is the cardinality of set of the subsets of {1, . . . , n} with k elements.
Definition 2.4. Let i = (i1 < i2 < . . . < ik) with ij ∈ {1, . . . , n}, then the Schubert cell Ci is
Ci = {U ∈ Gr(k, n) | pos(U) = i}
= {U ∈ Gr(k, n) | dim(U ∩ span{e1, . . . , er}) = j, ∀ij ≤ r ≤ ij+1 − 1}, where i0 := 0 and ik+1 = n by convention.
Example 2.4. The matrices in Mat4×2 whose position is (1, 3) are all of the form 1 0 0 x , 0 1 0 0 where x ∈ C. From the first definition of Schubert variety, it follows that U ∈ C(1,3) if and only if e1 ∈ U, e2 6∈ U and U ⊂ span{e1, e2, e3}. This is equivalent to requiring U ∩ Ce1 = Ce1, U ∩ span{e1, e2} = Ce1, U ∩ span{e1, e2, e3} = U, U ∩ span{e1, e2, e3, e4} = U. Equivalently,
dim(U ∩ Ce1) = 1, dim(U ∩ span{e1, e2}) = 1,
dim(U ∩ span{e1, e2, e3}) = 2, dim(U ∩ span{e1, e2, e3}) = 2.
This is in fact the second definition of C(1,3).
Exercise 2.1. Show that the Schubert cells C(1,3) and C(2,4) inside the Grass- mann variety Gr(2, 4) are affine varieties (i.e. find the set of polynomial equa- tions that they satisfy). Exercise 2.2. For any 2-subset i of {1, 2, 3, 4} find an element in the corre- sponding Schubert cell Ci. F Observe that Gr(k, n) = Ci, where i varies on the set of cardinality k subsets of {1, . . . , n}. 5
2.4. Projective variety structure. As shortly mentioned before, Gr(k, n) is a projective variety. The way of embedding it into a projective space goes as follows. First of all, fix a total ordering on the set of k-subsets of {1, . . . n} and then for any i as before and any n × k matrix M ∈ Matn×k we denote by pi(M) the determinant of the k × k-minor formed by the rows indexed by (n) i. We can first define a map Pk,n : Matn×k → C k given by M 7→ (pi(M)). Then we notice that if M = MU for some U ∈ Gr(k, n), at least one entry of Pk,n(M) is 6= 0, since U has dimension k and hence M has rank k. Moreover, we observe that if we perform column operations and hence get M 0 from M (or, n equivalently, we choose another basis for U) the k -tuple is rescaled by a global 0 (non-zero) factor, i.e. there exists a λ 6= 0 such that Pk,n(M ) = λPk,n(M). We conclude that we have a well-defined (injective) map (n)−1 P : Gr(k, n) ,→ k ,U 7→ (p (M )). k,n PC i U Such a map is called Pl¨uckerembedding. It can be shown that the image is the zero locus of certain quadratic (homogeneous) polynomials. Exercise 2.3. Determine the image under the Pl¨uckerembedding of the space U from (3). Exercise 2.4. For any 2-subset i of {1, 2, 3, 4} find an element in the corre- sponding Schubert cell Ci. 2.5. Schubert varieties.
Definition 2.5. Let i = (i1 < i2 < . . . < ik) with ij ∈ {1, . . . , n}. The Schubert variety Xi is
Xi = {U ∈ Gr(k, n) | dim(U ∩ span{e1, . . . , eij }) ≥ j}
(n)−1 Via the Pl¨ucker embedding, the (Zariski) topology on P k induces a topol- ogy on Gr(k, n). With respect to this topology, Xi is the closure of Ci. It is clear from the definition that G Xi = Cl for appropriate l.
Example 2.5. Let us assume n = 4, k = 2 and i = (1, 3). Then U ∈ X(1,3) if and only if
dim(U ∩ Ce1) ≥ 1, and dim(U ∩ span{e1, e2, e3}) ≥ 2. We see immediately that the only freedom we have is for the dimension of the intersection of U with span{e1, e2}, which can be 1 or 2. We hence conclude that
X(1,3) = C(1,2) t C(1,3).
Exercise 2.5. For which l (2-subsets of {1, 2, 3, 4}) is Cl contained in X(2,4)? Let i, l be two (increasingly ordered) k subsets of {1, . . . , n}. What is the sufficient and necessary condition such that Cl ⊆ Xi? 6 MARTINA LANINI
Example 2.6. Let us assume n = 4, k = 2 and i = (2, 4). Then
X(2,4) = {U ∈ Gr(2, 4) | dim(U ∩ span{e1, e2}) ≥ 1, dim(U ∩ span{e1, e2, e3, e4}) ≥ 2}
= {U ∈ Gr(2, 4) | dim(U ∩ span{e1, e2}) ≥ 1},
since {U ∈ Gr(2, 4) | dim(U ∩ span{e1, e2, e3, e4}) ≥ 2} = Gr(2, 4). Now, recall Yi from (1). Clearly, U ∩ Wi 6= 0 if and only if dim(U ∩ Wi) ≥ 1. We conclude that Yi ' C(2,4) n for any i (where the isomorphism is obtained by any automorphism of C which sends a basis of Wi to span{e1, e2}). We can define a partial order relation on the set of k subsets of {1, . . . , n} by
l ≤ i ⇔ Cl ⊆ Xi Definition 2.6. Such a partial order is called the Bruhat order. Exercise 2.6. Determine the poset structure (i.e. draw the Hasse diagram3) of the 2 subsets of {1,... 4}. Can you find some examples of pairs (k, n) such that the partial order we described before is actually a total order?
3. The case of the flag variety n Definition 3.1. A complete flag V• = (Vi)i=1,...n−1 in C is a sequence of n nested subspaces of C n {0} ⊂ V1 ⊂ V2 ⊂ ... ⊂ Vn−1 ⊂ C . n We denote by Fln the set of (complete) flags inside C .
3.1. A canonical representative in Matn×n. After dealing with the Grass- mannian it is not hard to believe that Fln is a projective variety. As before, to describe the embedding into a projective space we will use some matrix representation. n Given a complete flag V• = (Vk)k=1,...,n−1, we can construct a basis of C inductively by starting with a generator b1 for V1 and hence completing the basis (b1, . . . bk) of Vi to a basis (b1, . . . , bk, bk+1) of Vk+1 for k ≥ 1. We can hence arrange the basis vectors to form an n × n matrix (whose columns are the bk’s, written in terms of the standard basis (e1, . . . en)). Finally, we put the matrix into column echelon form, keeping in mind that this time not all column operations are allowed: X can substitute (column k) only by ar · (column r), ar ∈ C and ak 6= 0. r≤k
We denote the resulting matrix by MV• and by MVk the n×k-matrix obtained by considering the first k columns of MV• only.
3Given a poset (P, ≤) its Hasse diagram is an oriented graph whose vertices are the elements of the poset and there is an edge x → y if and only if x < y and if there exists an element z ∈ P such that x ≤ z ≤ y, then either z = x or z = y. 7
Example 3.1. Let n = 3, and consider the flag V• given by 3 {0} ⊂ C(2e2 + e3) ⊂ span{e2 + 2e3, 3e1 + e2 + e3} ⊂ C . We can pick 0 3 1 b1 = 1 , b2 = 1 , b3 = 0 . 2 1 0 Then, 0 3 1 0 3 1 0 3 1 0 61 1 1 1 1 1 1 0 → 2 1 0 → 2 2 0 → 2 10 = MV• . 2 1 0 1 1 0 1 0 0 100 3.2. Projective algebraic variety structure. By the previous discussion, we get an embedding of Fln into a product of projective spaces: (n)−1 (n)−1 ( n )−1 P : Fl ,→ 1 × 2 × ... × n−1 n PC PC PC V• 7→ ((pi1 (MV1 )) , (pi2 (MV2 )) ,..., (pin−1 (MVn−1 )), where ik varies in the set of k-subsets of {1, . . . , n}. The image of this map (which is still called Pl¨uckerembedding) is the zero locus of certain (homogeneous) quadratic polynomials also in this case.
3.3. Schubert cells. We want now to give the definition of Schubert cell, and hence of Schubert variety, in this setting. Let us get back to our matrix representation MV• of the flag V•. We observe that the pivotal ones form a permutation σ ∈ Sn (defined as σ(k) = j if the pivotal 1 of the k-th column is on the j-th row).
Definition 3.2. We call this permutation the position of the flag V•, and denote it by pos(V•).
Example 3.2. The position of the flag V• from Example 3.1 is the permutation σ ∈ S3 given by σ(1) = 3, σ(2) = 2, σ(3) = 1, which in the 2-line notation is represented by 1 2 3 σ = . 3 2 1
4 Exercise 3.1. Consider the following basis of C : 8 5 7 6 4 2 0 0 b1 = , b2 = , b3 = , b4 = . 2 1 3 0 0 0 4 4
What is the position of the flag V•, where Vi := spanC{b1, . . . , bi}?
Definition 3.3. For a permutation σ ∈ Sn, the Schubert cell Cσ is defined as
Cσ := {V• | pos(V•) = σ}. 8 MARTINA LANINI
We have that Fl = F C . n σ∈Sn σ As in the Grassmannian case, we want now to give an equivalent definition of the Schubert cell Cσ. We define Ei := spanC{e1, e2, . . . , ei} and call E• := (Ei)i=1,...,n−1 the stan- dard flag. We have an alternative description of Cσ as the set of flags in a given relationship (relative position) with the standard flag: Cσ = V ∈ Fln | dim(Vk ∩ Ei) = #{r | r ≤ k and σ(r) ≤ i} ∀1 ≤ k, i ≤ n − 1 . Exercise 3.2. Verify that if we only consider the k-th piece of any flag, we obtain a Schubert cell in Gr(k, n).
Exercise 3.3. For any permutation σ ∈ S3 exhibit an element of the corre- sponding Schubert cell Cσ.
For a permutation σ ∈ Sn, the Schubert variety Xσ is again the (Zariski) closure of the corresponding cell, which can be described explicitely as Xσ = V ∈ Fln | dim(Vk ∩ Ei) ≥ #{r | r ≤ k and σ(r) ≤ i} ∀1 ≤ k, i ≤ n − 1 . F Also in this case, Xσ = Cτ , for appropriate τ. We equip the symmetric group with the partial order ≤ defined as
τ ≤ σ ⇔ Cτ ⊆ Xσ. Definition 3.4. This is the Bruhat order ≤ on the symmetric group.
Exercise 3.4. Describe the partial order ≤ on S3. Let σ, τ ∈ Sn. Can you give a necessary and sufficient condition for τ ≤ σ to hold? We conclude by giving a combinatorial description of the Bruhat order on the symmetric group. In order to do that, we recall that for any permutation σ ∈ Sn its inversion set is defined as Inv(σ) = {(i, j) | i < j and σ(i) > σ(j)}. 1 2 3 Example 3.3. Let σ = . Then Inv(σ) = {(1, 2), (1, 3), (2, 3)} 3 2 1 Thus we have the following, powerful, characterisation of the Bruhat order:
Theorem 3.1. Let σ, τ ∈ Sn. Then σ ≤ τ in the Bruhat order if and only if there exists a sequence of transpositions t1, . . . tr ∈ Sn such that τ = trtr−1 . . . t1σ and if we set σ0 := σ, σi := ti . . . t1σ (i = 1, . . . r), then
#Inv(σi) < #Inv(σi+1) for all i = 0, . . . r − 1.
Exercise 3.5. Let GLn be the group of invertible n×n-matrices with coefficients in C. Then GLn acts on Fln: take a flag V• and construct inductively a basis n 0 (b1, . . . bn) of C as before, then for any g ∈ GLn we let g · V• be the flag V•, where 0 Vi := spanC{gb1, . . . , gbi}. (Clearly, gbi is the column vector obtained by matrix multiplying g and bi). 9
(1) Denote by B the subgrop of GLn consisting of the upper triangular ma- trices: A = (aij) ∈ B if and only if aij = 0 for any 1 ≤ j < i ≤ n and aii 6= 0 for any 1 ≤ i ≤ n. Use the Orbit-Stabiliser Theorem to show that there is a bijection
GLn/B ↔ Fln.
(2) Can you also find a subgroup P ≤ GLn such that there is a bijection GLn/P ↔ Gr(k, n)? (3) Let T denote the subgroup of GLn consisting of diagonal matrices. The action of GLn restricts to an action of T on Fln. Describe the fixed point set.
References [Fu] W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry, Cambridge University Press 1997. [Gi] M. Gillespie, Variations on a Theme of Schubert Calculus, preprint 2018, 1804.08164. [Schu] H. Schubert, Kalk¨ulder abz¨ahlendeGeometrie, Teubner Verlag, Leipzig, 1879.