A notion of inversion number associated to certain quiver flag varieties

We define an algebraic variety X(d,A) consisting of matrices whose rows and columns are partial flags. This is a smooth, projective variety, and we describe it as an iterated bundle of Grassmannian varieties. Moreover, we show that X(d,A) has a cell decomposition, in which the cells are parametrized by certain matrices of sets and their dimensions are given by a notion of inversion number. On the other hand, we consider the Spaltenstein variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are affine bundles over certain varieties called YT , parametrized by semistandard tableaux T . We show that the varieties YT are in fact isomorphic to varieties of the form X(d,A). We deduce that each variety YT has a cell decomposition, in which the cells are parametrized by certain row-increasing tableaux obtained by permuting the entries in the columns of T and their dimensions are given by the inversion number recently defined by P. Drube for such row-increasing tableaux. Mathematics Subject Classifications: 05A05, 05A19, 14M15


Introduction
Given the following data: • a p × q matrix of nonnegative integers d = (d i,j ) which is nondecreasing along the rows and the columns, i.e., d i,j d i ,j whenever i i , j j , • a chain of C-vector spaces A = (A 1 ⊂ . . .⊂ A q ) such that dim A j = d p,j , we define X(d, A) as the set of p × q matrices of vectors spaces V = (V i,j ) such that V i,j is a d i,j -dimensional subspace of A j for all i, j, (1) V i,j ⊂ V i ,j whenever i i , j j . (2) Thus X(d, A) consists of representations of the p × q rectangular quiver in the subcategory of vector spaces where we retain only inclusion morphisms.Clearly X(d, A) is a closed subset of the projective variety i,j where Grass k (H) stands for the Grassmannian variety of k-dimensional subspaces of a vector space H.
We outline some general facts on the variety X(d, A), which are explained in more detail in the rest of the paper: 1) X(d, A) is an iterated bundle of base type the following collection of Grassmannian varieties {Grass d i−1,j −d i−1,j−1 (C d i,j −d i−1,j−1 ) : 2 i p, 1 j q} (where d i,0 = 0), in particular X(d, A) is smooth, irreducible, and its Poincaré polynomial is explicitly determined; see Theorem 1.
2) For certain dimension matrices d, X(d, A) is a resolution of a Schubert variety in a natural way; see Remark 5 (a).In fact, in the case where the chain A is maximal, X(d, A) is a Bott-Samelson variety of special type; see Remark 5 (b).The definition of X(d, A) is related to the combinatorial construction of Bott-Samelson varieties given in [10]; see Remark 5 (c).
3) We define W = W(d) as the set of p × q matrices of sets ω = (ω i,j ) such that ω i,j is a subset of {1, . . ., d p,j } of cardinality d i,j for all i, j, (3) ω i,j ⊂ ω i ,j whenever i i , j j . ( In Section 3, we introduce a notion of inversion number n inv (ω) for the elements of W, and we show that the elements of W parametrize a cell decomposition X(d, A) = ω∈W C(ω) such that dim C(ω) = n inv (ω).
4) Our main original motivation in considering the variety X(d, A) is the study of Spaltenstein varieties.A Spaltenstein variety Fl k,u is a variety of partial flags (for dimension vector k) which are preserved by a given nilpotent endomorphism u : C n → C n (see Section 4.1).This variety is in general not irreducible.As it is recalled in Section 4.2, there is a natural partition of Fl k,u into locally closed subsets Fl k,u = T ∈STab k (λ(u)) Fl k,u,T parametrized by semistandard tableaux T whose shape is the Jordan form of u (seen as a Young diagram).Moreover, the closures Fl k,u,T are the irreducible components of Fl k,u .In Proposition 17, we show that for each subvariety Fl k,u,T , there is an affine bundle where Y T is a certain projective variety (realized as the subvariety of Fl k,u,T formed by flags which are homogeneous with respect to a grading adapted to the filtration C n = q j=1 ker u j ).The main results of Section 4 concern the structure of the variety Y T .In Theorem 20 we show that Y T is isomorphic to a variety of the form X(d, A).
In Theorem 27, relying on Section 3, we then show that Y T has a cell decomposition parametrized by the set RTab(T ) of all row-increasing tableaux obtained by permuting entries in the columns of T , moreover the cell decomposition is such that the dimensions of the cells Y (τ ) coincide with the inversion numbers n inv (τ ) defined by P. Drube [4] for such row-increasing tableaux.
Spaltenstein varieties are considered in [12,13].Computations of their Poincaré polynomials can also be deduced from [3,11].In the present paper, we are able to provide closed formulas for the Poincaré polynomials of the varieties Y T and the Spaltenstein variety Fl k,u (see Corollaries 21, 29, and Remark 30).This generalizes similar results obtained for Springer fibers in [6].The results obtained in Section 4 also give a geometric interpretation of the recent results of [4,5].

Notation and mathematical background
All the algebraic and geometric constructions are made over C. By |M | we denote the cardinality of a set M .For a positive integer k, we consider the polynomials [k] x := 1 + x + . . .+ x k−1 and [k] x !:= [1] Given an algebraic variety Y , we denote by H i (Y, Q) its cohomology spaces (considering sheaf cohomology with rational coefficients) and by H i c (Y, Q) its cohomology the electronic journal of combinatorics 25(3) (2018), #P3.41 with compact support.Note that H i (Y, Q) = H i c (Y, Q) whenever Y is projective.Let P (Y )(t) := i 0 dim H i (Y, Q) t i be the Poincaré polynomial.
In fact, all varieties considered in this paper satisfy the parity vanishing condition H i (Y, Q) = 0 whenever i is odd.Hence we may renormalize the Poincaré polynomial as P (Y )(x) = i 0 dim H 2i (Y, Q) x i , setting x = t 2 .
A sufficient condition for a projective variety Y to have this parity vanishing condition is the existence of a cell decomposition.By cell decomposition, we mean here a partition into finitely many locally closed subsets that can be numbered as Y = Y 1 . . .Y k so that Y 1 . . .Y is closed for all and each Y is isomorphic to an affine space A dim Y .Then For example, the decomposition x .More generally, letting B ⊂ GL n (C) be the subgroup of upper triangular matrices, by the Bruhat decomposition, the partition into B-orbits of any variety of partial flags of C n is a cell decomposition.In the case of the Grassmannian variety Grass k (C n ), the cells are parametrized by the subsets I ⊂ {1, . . ., n} with k elements: the cell C(I) is the B-orbit of the subspace If X, Y, F are projective varieties, Y, F are smooth, irreducible, satisfy the aforementioned parity vanishing condition, and ϕ : X → Y is a locally trivial fiber bundle with fiber isomorphic to F , then X is smooth, irreducible, satisfies the parity vanishing condition, and P (X)(x) = P (Y )(x) • P (F )(x).
The notion of iterated bundle is defined by induction.An iterated bundle of base type {Y 1 } is a variety isomorphic to Y 1 ; for k 2, we say that X is an iterated bundle of base type {Y 1 , . . ., Y k } if (up to renumbering the Y 's), there is a locally trivial fiber bundle X → Y k whose typical fiber is an iterated bundle of base type {Y 1 , . . ., Y k−1 }.Assume that X, Y 1 , . . ., Y k are projective.If Y 1 , . . ., Y k are smooth, irreducible, satisying the parity vanishing condition, then so is X, and P (X)(x) = k =1 P (Y )(x).For instance the variety of complete flags Fl(C n ) is an iterated bundle of base type 2 Structure of the variety X(d, A) The variety X(d, A) is endowed with a natural action of the group which is a parabolic subgroup of GL(A q ).Given nonnegative integers a b c, we denote Whenever V a ⊂ V c are vector spaces of dimensions a and c, respectively, the variety of b-dimensional spaces H such that Theorem 1.The variety X(d, A) is an iterated bundle of base type the sequence of Grassmannian varieties its cohomology spaces H m (X(d, A), Q) vanish in odd degrees, and its Poincaré polynomial is given by (a) Assume that d p,j = d p−1,j for all j ∈ {1, . . ., q}.Then, X(d, A) ∼ = X( d, A), where d = ( di,j ) is the (p − 1) × q matrix given by di,j = d i,j for all (i, j) ∈ {1, . . ., p − 1} × {1, . . ., q}.
In the situation of Lemma 3 (b), we have a canonical isomorphism It remains to show that ϕ is trivial over the B-orbit of H whenever B is a Borel subgroup of GL(A j 0 /A j 0 −1 ) (this fact guarantees that ϕ is locally trivial, since there is a Borel subgroup B for which the orbit B • H is open).By the properties of Schubert cells (see, e.g., [2]), there is a unipotent subgroup U ⊂ B such that the map is an isomorphism of algebraic varieties.Moreover there is a natural embedding of GL(A j 0 /A j 0 −1 ) into P A , which yields an action of GL(A j 0 /A j 0 −1 ) on X(d, A), such that the map ϕ is GL(A j 0 /A j 0 −1 )-equivariant.Whence a commutative diagram where pr 1 is the projection on the first factor while the isomorphism ξ is given by ξ Proof of Theorem 1.The proof is done by induction on the tuple (p, d p,1 , . . ., d p,q ) (considering lexicographic order), with immediate initialization for p = 1 (in which case X(d, A) is reduced to a point).The induction step is yielded by Lemma 3.
Remark 4. If G is an algebraic group and G is a closed subgroup acting on an algebraic variety Y , then we let G act on G×Y by g •(g, y) = (gg −1 , g y) and denote by G× G Y := (G×Y )/G the quotient variety.The latter variety is equipped with a G-action in a natural way.Lemma 3 (b) yields the following inductive formula, in terms of a P A -equivariant isomorphism of varieties X(d, A) where H is any d p−1,j 0 -dimensional space such that A j 0 −1 ⊂ H ⊂ A j 0 .As before P A ⊂ GL(A q ) is the parabolic subgroup of elements which fix the partial flag A = (A 1 ⊂ . . .⊂ A q ), while by P A,H we denote the (parabolic) subgroup P A,H = {g ∈ P A : g(H) = H}.
Remark 5. (a) We consider the space C n (n 1) and its standard basis (ε 1 , . . ., ε n ), and let A = (A 1 , . . ., A q ) be a standard partial flag, i.e., A j = ε s : 1 s j C , for some 1 1 < 2 < . . .< q = n.Thus P A is a standard parabolic subgroup.Given a sequence of positive integers k = (k 1 < . . .< k p = n), let Fl k (C n ) be the variety of partial flags F = (F 1 ⊂ . . .⊂ F p = C n ) with dim F i = k i for all i.A permutation w ∈ S n gives rise to the element Let d w = (d w i,j ) be the p × q matrix given by d w i,j := |{w 1 , w 2 , . . ., w k i } ∩ {1, 2, . . ., j }|.
The P A -orbit of F w is given by and its closure is the Schubert variety Then, the map is a resolution of singularities of the Schubert variety P A • F w (this map is proper since X(d w , A) is projective; it is birational since its restriction over P A • F w is an isomorphism; finally it follows from Theorem 1 that the variety X(d w , A) is smooth).(b) Now assume that j = k j = j for all j, hence Fl( is the standard complete flag, and B := P A is the Borel subgroup of upper triangular matrices.For a permutation w ∈ S n , the matrix d w = (d w i,j ) 1 i,j n is here given by d w i,j = |{w 1 , . . ., w i } ∩ {1, . . ., j}| for all i, j ∈ {1, . . ., n}.
Then, the variety X(d w , A) is (B-equivariently) isomorphic to the Bott-Samelson variety Z [w] associated to the reduced decomposition [w] just defined.This fact can be shown directly by induction, by relying on the inductive formula given in Remark 4. It also follows from part (c) of the present Remark.(c) Following the terminology of [10], a subset family is a collection D of subsets of {1, . . ., n}.A flagged representation of D is a sequence of subspaces (V C ) C∈D of C n such that dim V C = |C| and V C ⊂ V C whenever C ⊂ C .In fact, we focus on subset families such that {1, . . ., j} ∈ D for all j ∈ {1, . . ., n}, and on flagged representations (V C ) such that V {1,...,j} = A j for all j.Given a subset family D, let I B D be the set of all such flagged representations; it is a projective variety endowed with a natural action of B.
Given a permutation w ∈ S n , let D w be the subset family Then, there is a natural isomorphism , where D + i is the subset family then it is easy to show (by induction) that both subset families D w and D

Discrete data and inversion number
In this section we set n := d p,q (= dim A q ) and fix a basis (ε 1 , . . ., ε n ) of the space A q such that A j = ε a : a ∈ {1, . . ., d p,j } C for all j ∈ {1, . . ., q}.
(a) Let a, b ∈ {1, . . ., n} such that a = b.For j ∈ {1, . . ., q}, we write a < j b if one of the following two conditions is fulfilled: • a appears before b in the filtration (7), i.e., there is i ∈ {1, . . ., p} such that a ∈ ω i,j and b / ∈ ω i,j ; or • b does not appear in the filtration (7) and is greater than a, i.e., b / ∈ ω p,j (that is, b > d p,j ) and b > a.

Note that Inv
the electronic journal of combinatorics 25(3) (2018), #P3.41 Recall that P A ⊂ GL(A q ) is the parabolic subgroup of elements which preserve the partial flag A = (A 1 , . . ., A q ).Hence (a) The elements V ω , for ω ∈ W(d), are exactly the S A -fixed points of the variety X(d, A).
The proof relies on a discrete version of Lemma 3 (b).
• Let d = ( d i,j ) be the p × q matrix as in Lemma 3 (b).
Set I = {(a, b) ∈ {1, . . ., n} 2 : a < b} and let σ : I → I be the bijection defined by By definition of n inv (ω) and n inv ( ω), for proving (8), it suffices to check that for every couple (a, b) To this end, we need to compare the relative positions of (a, b) and (σ(a), σ(b)) in each column of ω and ω, respectively.We make two observations.The first one follows from the construction of ω: The second one follows from the construction of σ: Thus in these two cases, (13) holds also for j = j 0 .At this stage, because of (13), we obtain that every couple (a, b) 0 } fulfills the equivalence in (9).Next we consider a couple (a, b) ∈ J 0 .Thus a, b ∈ ω c 0 = ω p,j 0 \ ω p−1,j 0 , and this yields a ∼ j 0 b.For every j < j 0 , we have b > a > d p−1,j 0 d p,j hence a < j b.On the other hand by ( 11)- (12) we have σ(b) > σ(a) and σ(b) / ∈ ω p,j 0 , i.e., σ(b) > d p,j 0 d p,j for all j j 0 , whence σ(a) < j σ(b) for all j ∈ {0, . . ., j 0 }.Altogether we conclude that (a, b) / ∈ Inv j (ω) and σ(a, b) / ∈ Inv j ( ω) for all j ∈ {1, . . ., j 0 }, and that the relative positions of (a, b) and σ(a, b) before the (j 0 + 1)-th column of ω and ω, respectively, are both equal to <.These facts, combined with (13), guarantee that every couple (a, b) ∈ J 0 fulfills the equivalence in (9).
Finally we consider a couple (a, b) ) in view of ( 12).This yields b ∈ ω p−1,j 0 (= ω 0 ) and a / ∈ ω p−1,j 0 , hence b < j 0 a, while for every j < j 0 we have b > a > d p,j 0 −1 d p,j hence a < j b.On the other hand, by (11) we get σ(b) ∈ ω p−1,j 0 and σ(a) / ∈ ω p−1,j 0 , hence σ(b) < j 0 σ(a), and for every j < j 0 we get σ(a) > σ(b) and σ(a) > d p,j 0 −1 d p,j hence σ(b) < j σ(a).Altogether this implies that (a, b) / ∈ Inv j (ω) whenever j < j 0 and (a, b) ∈ Inv j 0 (ω) on one hand, σ(a, b) / ∈ Inv j ( ω) whenever j j 0 on the other hand; in addition the relative positions of the couples (a, b) and (σ(a), σ(b)) before the (j 0 + 1)-th column of ω and ω, respectively, are both equal to >.For j > j 0 we have a, b ∈ ω p,j 0 ⊂ ω p,j .On the basis of (11) we deduce that the relative positions of (a, b) and (σ(a), σ(b)) in the j-th column of ω and ω, respectively, coincide.Hence, for j > j 0 , we have (a, b) ∈ Inv j (ω) if and only if (σ(b), σ(a)) ∈ Inv j ( ω).Therefore the couple (a, b) fulfills the equivalences in ( 9) and (10).The proof of the lemma is complete.d, A) is fixed by S A if and only if each subspace V i,j is S A -stable, which means that V i,j is a sum of S A -eigenspaces, i.e., is of the form V i,j = ε a : a ∈ ω i,j C for some subset ω i,j ⊂ {1, . . ., n}.This subset must be of cardinality dim V i,j = d i,j and the inclusion V i,j ⊂ V i ,j yields ω i,j ⊂ ω i ,j whenever i i , j j ; in addition, the equality V p,j = A j yields ω p,j = {1, . . ., d p,j } for all j.Hence ω := (ω i,j ) is an element of W(d) and we have V = V ω .
(b) The proof is done by induction on the tuple (p, d p,1 , . . ., d p,q ) (considering lexicographic order) with immediate initialization if p = 1.So assume that p 2 and let us distinguish two cases, as in the statement of Lemma 3. Case 1: d p,j = d p−1,j for all j ∈ {1, . . ., q}.
Let Y be the variety of d p−1,j 0 -dimensional subspaces H such that A j 0 −1 ⊂ H ⊂ A j 0 and let us consider the map as in Lemma 3.For ω 0 ∈ W 0 , with W 0 as in Lemma 10, let Denoting by B ⊂ GL(A q ) the subgroup of automorphisms which are upper triangular in the basis (ε 1 , . . ., ε n ), we have a cell decomposition where J 0 is the set given in Lemma 10 (see ( 6)).Note that (14) yields a partition , with φ as in Lemma 10.Hence, for showing Theorem 9 (b), given any ω 0 ∈ W 0 , it suffices to construct a cell decomposition which satisfies conditions (i) and (ii) of the statement.Letting d = ( d i,j ) and A = ( A j ) be as in Lemma 3 (b) (for H = H ω 0 ), we get by Lemma 3 (and its proof) a trivialization of ) and an isomorphism Letting ε a := ε σ −1 (a) (for a = 1, . . ., n), with σ as in Lemma 10, we get a basis ( ε 1 , . . ., ε n ) of A q such that A j = ε a : 1 a d p,j C for all j = 1, . . ., q.By induction hypothesis, we have a cell decomposition Invoking Lemma 10, we deduce that this cell decomposition satisfies conditions (i) and (ii) of Theorem 9 (b).This completes the proof of the theorem.
Example 12. (a) In the special case where q = 1, the chain of subspaces A = (A 1 ) consists of a single space, say A 1 = C n , while the dimension matrix d = (d i,1 ) p i=1 consists of a single column.Then the variety X(d, A) coincides with the variety of partial flags . . .d 2,q = n) be the entries in the second row, i.e., the dimensions of the subspaces forming the (fixed) sequence A = (A 1 ⊂ . . .⊂ A q = C n ).In this case X(d, A) can be identified with the subvariety Y ⊂ Fl k (C n ) consisting of partial flags F = (F 1 ⊂ . . .⊂ F q ⊂ C n ) such that F j ⊂ A j for all j = 1, . . ., q.Note that Y is P A -stable, smooth, and irreducible (by Theorem 1), hence it is the closure of a P A -orbit, i.e., a (smooth) Schubert variety of Fl k (C n ).Theorem 9 retrieves the properties of the decomposition of this Schubert variety into Schubert cells.Specifically, the map {w ∈ S n : w({1, . . ., d 1,j }) ⊂ {1, . . ., d 2,j } ∀j = 1, . . ., q}/S k → W(d) given by ω 1,j := {w 1 , . . ., w d 1,j }, ω 2,j := {1, . . ., d 2,j } for all j = 1, . . ., q is a bijection such that, for ω ∈ W(d), the number n inv (ω) coincides with the length of the minimal representative of the corresponding coset wS k .

Application to Spaltenstein varieties
In this section we fix the following data: ) is an increasing sequence of integers.As before, we denote by Fl k (C n ) the variety of partial flags • u : C n → C n is a nilpotent endomorphism.

The Spaltenstein variety Fl k,u
The Spaltenstein variety Fl k,u is the subvariety of Fl k (C n ) defined by Thus Fl k,u is a closed subvariety of Fl k (C n ), hence a projective variety -provided that it is nonempty.Let λ(u) = (λ 1 . . .λ r ) be the partition of n formed by the sizes of the Jordan blocks of u.This partition can be represented by a Young diagram (also denoted λ(u)) of rows of lengths λ 1 , . . ., λ r .By λ(u) * = (λ * 1 . . .λ * λ 1 ) we denote the dual partition of n, i.e., the lengths of the columns of λ(u).
The dimension vector k yields a composition of n denoted µ(k . By µ(k) + we denote the partition of n obtained by putting the sequence µ(k) in nonincreasing order.
We emphasize the following properties of the Spaltenstein variety Fl k,u .
(a) Fl k,u is nonempty if and only if µ(k) + λ(u) * , where stands for the dominance order. .
(c) Moreover, there is a bijection between the set of irreducible components of Fl k,u and the set STab k (λ(u)) of semistandard tableaux of shape λ(u) and weight µ(k).
Recall that a semistandard tableau of shape λ(u) and weight µ(k) = (µ 1 , . . ., µ p ) (with ) is a numbering of the boxes of the Young diagram λ(u) by the integers 1, 2, . . ., p, comprising µ i boxes of number i for all i, such that the entries in each row are increasing from left to right and the entries in each column are nondecreasing from top to bottom.The set STab k (λ(u)) of such semistandard tableaux is nonempty precisely when the condition µ(k) + λ(u) * is fulfilled.
the electronic journal of combinatorics 25(3) (2018), #P3.41 In the next subsection, we recall from [12,13] an explicit parametrization of the components of Fl k,u by the semistandard tableaux of the set STab k (λ(u)).

4.2
The subvarieties Fl k,u,T and Y T := (Fl k,u,T ) S associated to semistandard tableaux Given F = (F 0 , . . ., F p ) ∈ Fl k,u , for each i, we get by restriction a nilpotent endomorphism u| F i : F i → (F i−1 ⊂)F i , whose Jordan form can be encoded by a partition/a Young diagram λ(u| F i ) k i .This yields a chain of Young diagrams Let T be the tableau of shape λ(u) obtained by putting the number i in the boxes of λ(u| ) contains at most one box, which guarantees that T is a semistandard tableau, in fact an element of STab k (λ(u)).
For every semistandard tableau T ∈ STab k (λ(u)), we define where T | i stands for the subtableau of T of entries i.The above discussion shows that the Spaltenstein variety Fl k,u is the disjoint union of the subsets Fl k,u,T so-obtained.In fact, we have the following result: Proposition 15 ( [12,13]).
where c j (T | i ) stands here for the number of entries i in the first j columns of T .This description shows that the subsets Fl k,u,T (for T ∈ STab k (λ(u))) coincide with the intersections between Fl k,u and the Q-orbits of Fl k (C n ).Since every Q-orbit of Fl k (C n ) is locally closed, this guarantees that the subsets Fl k,u,T are locally closed in Fl k,u .
Remark 16.The subspace n Q := {y ∈ End(C n ) : y(ker u j ) ⊂ ker u j−1 ∀j 1} is the nilradical associated to the parabolic subgroup Q.The nilpotent endomorphism u is a Richardson element of Q, in the sense that the orbit Q Any partial flag F = (F 0 , . . ., F p ) ∈ Fl k,u,T gives rise to a parabolic nilradical n(F . Then the smoothness of Fl k,u,T (stated in Proposition 15) also follows from the fact that (Q • u) ∩ n(F ) is a smooth variety (since it is open in the space n Q ∩ n(F )) while g → g(F ) and π are smooth maps.

The variety Y T
For deducing more facts on the structure of the subvariety Fl k,u,T , we need more notation.Since λ(u) = (λ 1 , . . ., λ r ) is the Jordan form of u, there is a basis (ε i,j : 1 i r, 1 j λ i ) of the space C n such that For j ∈ {1, . . ., λ 1 }, we set Thus we get a grading moreover the subspaces K j satisfy ker u j = K 1 ⊕ . . .⊕ K j = ker u j−1 ⊕ K j and u(K j ) ⊂ K j−1 for all j ∈ {1, . . ., λ 1 } (with K 0 := 0).Let S = {h(t) : t ∈ C * } ⊂ GL n (C) be the rank-one subtorus such that h(t)v = t −2j v for all v ∈ K j , for all j = 1, . . ., λ 1 .
Thus h(t)uh(t) −1 = t 2 u for all t ∈ C * , hence each element of S normalizes u, and so stabilizes the kernels ker u j .This implies that S acts on the Spaltenstein variety Fl k,u and preserves the subvariety Fl k,u,T for every semistandard tableau T ∈ STab k (λ(u)).We can therefore define In other words, Y T is the subset of flags F = (F 0 , . . ., F p ) ∈ Fl k,u,T whose subspaces F i are homogeneous with respect to the grading of (16) in the sense that The notation Y T is not ambiguous since, up to isomorphism, the variety Y T only depends on the semistandard tableau T .
Proposition 17. Y T is a smooth, projective, and irreducible variety.Moreover, the map ϕ T : Fl k,u,T → Y T , F → lim t→0 h(t)F is an algebraic affine bundle.
Proof.Our aim is to apply [7, Proposition 2].The torus S = {h(t) : t ∈ C * } acts by conjugation on the Lie algebra gl n (C) = End(C n ), and this action induces a grading Note that: • u ∈ g(2); • the Lie subalgebra g( 0) := i 0 g(i) consists of the endomorphisms y : C n → C n which preserve the kernels ker u j (j = 1, . . ., λ 1 ), in particular every endomorphism which commutes with u belongs to g( 0).
These observations mean that the grading gl n (C) = i∈Z g(i) is good for u in the sense of [7,Proposition 2].Moreover the second observation means that the parabolic subgroup Q ⊂ GL n (C) formed by the elements which preserve the kernels ker u j is corresponding to the cocharacter t → h(t) in the sense of [7, Section 2.1.3].As shown above, Fl k,u,T is the intersection between the Spaltenstein variety Fl k,u and a Q-orbit of the partial flag variety Fl k (C n ).We are now in position to apply [7, Proposition 2], which shows that Y T is smooth, projective, and that the map ϕ T is an algebraic affine bundle over each connected component of Y T .Therefore the proof of the proposition is complete once we know that Y T is also an irreducible variety.This fact is shown in Section 4.3 below (it follows from Theorem 1 and the claim made in the title of Section 4.3 below).
Remark 18.The reasoning made in [9,Section 11.16] shows that, if C ⊂ Y T is a locally closed subset isomorphic to an affine space, then so is its inverse image ϕ −1 T (C) ⊂ Fl k,u,T ⊂ Fl k,u (and the codimension of ϕ −1 T (C) in Fl k,u coincides with the codimension of C in Y T ).In Section 4.4, we show that the variety Y T has a cell decomposition for all semistandard tableau T ∈ STab k (λ(u)).By collecting the inverse images of these cells by the various maps ϕ T , we therefore obtain a cell decomposition of the whole Spaltenstein variety Fl k,u .

The variety Y T is isomorphic to a variety of the form X(d, A)
As in Section 4.2, we consider a nilpotent endomorphism u : C n → C n of Jordan form λ(u) = (λ 1 , . . ., λ r ) n. Thus q := λ 1 is the nilpotency order of u, i.e., u q = 0, u q−1 = 0.As in Section 4.2, we consider a grading such that ker u j = K 1 ⊕ . . .⊕ K j and u(K j ) ⊂ K j−1 for all j ∈ {1, . . ., q} (with K 0 := 0).We fix a semistandard tableau T ∈ STab k (λ(u)) and focus on the variety Y T = (Fl k,u,T ) S the electronic journal of combinatorics 25(3) (2018), #P3.41 of partial flags F = (F 0 , . . ., F p ) which both belong to the subvariety Fl k,u,T ⊂ Fl k,u and are homogeneous with respect to the grading C n = q j=1 K j , i.e., The tableau T has q columns and its entries belong to {1, . . ., p}.For i ∈ {1, . . ., p} and j ∈ {1, . . ., q}, we denote by c j (T | i ) (resp., c j (T | i )) the number of boxes in the j-th column (resp., in the first j columns) of the subtableau T | i ; i.e., the number of entries i in the j-th column (resp., in the first j columns) of T .
Proof.Assume that (i) holds.Hence F belongs to Fl k,u,T .In particular F belongs to Fl k,u , which implies that u(F i ) ⊂ F i−1 for all i ∈ {1, . . ., p}.Since u(K j ) ⊂ K j−1 , we deduce that u(K j ∩ F i ) ⊂ K j−1 ∩ F i−1 .In addition by ( 15) and the homogeneity of F , we have dim This shows (ii).Conversely assume that (ii) holds.Since F is already assumed to be homogeneous, we just need to show that F belongs to Fl k,u,T .Again the homogeneity of F , combined with the assumption in (ii), implies that By (15), we conclude that F ∈ Fl k,u,T .
Notation.When A = (a i,j ) is a p × q matrix (whose coefficients a i,j are numbers, linear spaces, or sets), we define its shifting A as the (p + q − 1) × q matrix whose j-th column has the following content: (a 1,j , . . ., a 1,j j terms , a 2,j , . . ., a p−1,j , a p,j , . . ., a p,j q + 1 − j terms ).
• Let d T = (d i,j ) be the p × q matrix of nonnegative integers given by d i,j = c q+1−j (T | i ) (= the number of entries i in the (q + 1 − j)-th column of T ).
Then, there is an isomorphism of varieties Φ T : Y T → X(d T , A) given by F = (F 0 , . . ., F p ) → V where V = (V i,j ) is the p × q matrix of linear spaces such that V i,j = u q−j (F i ∩ ker u q+1−j ) for all i = 1, . . ., p, all j = 1, . . ., q.
Combining Theorems 1 and 20, we obtain in particular a closed formula for the Poincaré polynomial of the variety Y T : Corollary 21.Let d T = (d i,j ) be the p × q matrix of Theorem 20.Set by convention d i,0 := 0. Then: and (the two tableaux of Example 14), we get and Corollary 21 yields dim Proof of Theorem 20.We first check that the variety X(d T , A) is well defined.The tableau T being semistandard, every box of entry i contained in the j-th column of T (j 2) is on the right of a box of entry i − 1 (contained in the (j − 1)-th column).Thus we must have ) for all i = 1, . . ., p, all j = 2, . . ., q.
Relations ( 18) and ( 19) ensure that the shifted matrix d T has nondecreasing rows and columns.In addition the j-th subspace A j := u q−j (ker u q+1−j ) of the sequence A has dimension which coincides with the last coefficient of the j-th column of d T .These observations ensure that the variety X(d T , A) is well defined.Let ûj−1 : K j → A q−j+1 = u j−1 (ker u j ) denote the restriction of u j−1 .We note that ûj−1 is a linear isomorphism.Whenever M ⊂ C n is a subspace homogeneous with respect to the grading C n = q j=1 K j , we note that u j−1 (M ∩ ker u j ) = ûj−1 (M ∩ K j ).Both observations are used throughout the rest of the proof.
Next, we check that the map Φ T is well defined.So let F = (F 0 , . . ., F p ) ∈ Y T and let V = (V i,j ) be as in the statement.By Lemma 19, we have for all i = 2, . . ., p, all j = 1, . . ., q − 1. Next, it is clear that V i,j = u q−j (F i ∩ ker u q+1−j ) ⊂ u q−j (F i+1 ∩ ker u q+1−j ) = V i+1,j ⊂ V p,j = A j whenever i = 1, . . ., p − 1, j = 1, . . ., q.Finally, invoking again Lemma 19, we get for all i, j.These observations guarantee that the shifted matrix of spaces V belongs to the variety X(d T , A).
The well-defined map Φ T : Y T → X(d T , A) so-obtained is clearly algebraic.Assume that we know that Φ T is bijective.Then, since Y T and X(d T , A) are projective varieties, it is also bicontinuous.Since X(d T , A) is irreducible (by Theorem 1), we deduce that Y T is irreducible (which, by the way, completes the proof of Proposition 17).Since Y T and X(d T , A) are smooth varieties, by Zariski's main theorem (see, e.g., [1, §AG.18.2]),Φ T is in fact an isomorphism.Thus, the proof of the theorem is complete once we check that Φ T is bijective.
Let us check that Φ T is injective.So let In view of the definition of Φ T , this implies that u j−1 (F i ∩ ker u j ) = u j−1 (F i ∩ ker u j ) for all i, j, i.e., ûj−1 (F i ∩ K j ) = ûj−1 (F i ∩ K j ) with the notation ûj−1 the electronic journal of combinatorics 25(3) (2018), #P3.41 introduced above.Since ûj−1 is injective, we derive F i ∩ K j = F i ∩ K j for all i, j.Whence F i = F i for all i (since F, F satisfy (17)).
We further note that u(K 1 ∩ F i ) = 0 (since K 1 ⊂ ker u) and K j ∩ F 1 = 0 if j 2 (by (23) and the fact that the entry 1 appears only in the first column of the semistandard tableau T ).Whence, finally, u(K j ∩ F i ) ⊂ K j−1 ∩ F i−1 for all i = 1, . . ., p, all j = 1, . . ., q.
(24) By ( 23) and ( 24), F satisfies the conditions of Lemma 19 (ii).Therefore Lemma 19 guarantees that F ∈ Y T .Finally, by the first equality in (23), we have for all i, j.Hence V = V = Φ T (F ).The surjectivity of Φ T is established, the proof is complete.(a) Let RTab k (λ(u)) be the set of tableaux τ of shape λ(u) and entries 1, . . ., p, such that τ contains k i − k i−1 entries equal to i for all i, and the entries in each row of τ are (strictly) increasing from left to right.Note that STab k (λ(u)) is a subset of RTab k (λ(u)).
(b) Given a tableau τ ∈ RTab k (λ(u)), define its rectification Rect(τ ) to be the tableau of shape λ(u) obtained from τ by reordering the entries of each column in nondecreasing order from top to bottom.In fact, the tableau Rect(τ ) so-obtained is semistandard, hence belongs to STab k (λ(u)) (see [4]).Given a semistandard tableau T ∈ STab k (λ(u)), we define (c) In [4], P. Drube introduces a notion of inversion number n inv (τ ) which measures how far a row-increasing tableau τ ∈ RTab k (λ(u)) is from being semistandard.We summarize this definition here: an inversion pair of τ consists of two entries a < b in the same column of τ such that one of the following conditions holds -by (a 1 , a 2 , . . ., a ), resp., (b 1 , b 2 , . . ., b m ), we denote the (possibly empty) list of entries directly to the right of a, resp.b, in τ , read from left to right: • a is below b (in particular m) and a j = b j for all j = 1, . . ., ; or • there is j 0 min{ , m} such that a j = b j for 1 j < j 0 and a j 0 > b j 0 .
Recall that we have fixed a Jordan basis (ε ,j ) of u parametrized by the couples ( , j) with 1 r and 1 j λ (those couples correspond to the various positions of the boxes of the Young diagram λ(u)).The basis is such that u(ε ,j ) = ε ,j−1 if j 2 and u(ε ,1 ) = 0.
(a) The partial flag F τ belongs to the variety Y T for T = Rect(τ ).
(b) Let g : C n → C n be a linear isomorphism which is diagonal in the basis (ε ,j ), with n pairwise distinct eigenvalues, and such that gug −1 ∈ C * u.Such a g exists.Then g acts on Y T in a natural way and Proof.(a) By construction, dim F i is equal to the number of entries i in τ .Since τ belongs to the set RTab k (λ(u)), this number is equal to k i .Hence F ∈ Fl k (C n ).By construction, each subspace F i is spanned by vectors which belong to q j=1 K j , hence the flag F is homogeneous.Moreover for all i, j we have On the one hand this implies that dim F i ∩ K j = (number of entries i in the j-th column of τ ) = c j (T | i ) since the j-th columns of τ and T = Rect(τ ) have the same content.On the other hand, for j 2, using that the rows of τ are increasing, we get is an example of map g which fulfills the conditions; it satisfies g 0 ug −1 0 = 2 r u.The fact that g normalizes u and preserves each subspace K j guarantees that its natural action on partial flags stabilizes the variety Y T (see Lemma 19).The inclusion {F τ : τ ∈ RTab(T )} ⊂ (Y T ) g is clear.Conversely, if F = (F 0 , . . ., F p ) ∈ Y T is g-fixed, then each subspace F i is spanned by a family of eigenvectors (ε ,j : ( , j) ∈ I i ) for some subset I i of cardinality k i .Let σ be the tableau of shape λ(u) obtained by putting the number i in the ( , j) box of λ(u) whenever ( , j) ∈ I i \I i−1 .Since u(F i ) ⊂ F i−1 , the implication ( , j) ∈ I i ⇒ ( , j−1) ∈ I i−1 holds whenever i 1, j 2. This guarantees that the rows of σ are increasing, therefore the tableau σ belongs to RTab k (λ(u)), and F = F σ is the partial flag corresponding to this tableau in the sense of Definition 25.Finally, part (a) guarantees that Rect(σ) = T , i.e., σ ∈ RTab(T ).The proof is complete.
The main result of this section states as follows.
Theorem 27.We consider a semistandard tableau T ∈ STab k (λ(u)) and the corresponding variety Y T .There is a cell decomposition Y T = Y (τ ) parametrized by the row-increasing tableaux τ ∈ RTab(T ) (of rectification T ), which satisfies the following conditions: This result is a consequence of Theorems 9, 20 above and Proposition 28 below.Before stating the proposition, we review some notation.We fix a semistandard tableau T ∈ STab k (λ(u)).We consider the p × q matrix d T = (d i,j ) and the chain of subspaces A = (A 1 , . . ., A q ) := (u q−1 (ker u q ) ⊂ . . .⊂ u(ker u 2 ) ⊂ ker u) introduced in Theorem 20.The vectors ε a := ε a,1 (1 a r) form a basis of A q = ker u.Moreover A j = ε a : 1 a d p,j C for all j = 1, . . ., q, since A j = u q−j (K q+1−j ) and the vectors ε a,q+1−j (a = 1, . . ., d p,j ) generate K q+1−j .Finally, recall the set W(d T ) considered in Section 3. Every ω ∈ W(d T ) is a (p + q − 1) × q matrix of sets and gives rise to an element V ω ∈ X(d T , A) (see Definition 8).Recall the inversion number n inv (ω) defined in Definition 6.

Then the map
is a well-defined bijection.Moreover, this bijection satisfies where Φ T : Y T → X(d T , A) is the isomorphism of Theorem 20.
Case 1: Assume that i a < i b , i.e., a < j b.
In this case, the couple (i a , i b ) is an inversion of τ if and only if there is s 0 ∈ {1, . . ., m} such that (i a ) s = (i b ) s whenever 1 s < s 0 and (i a ) s 0 > (i b ) s 0 .Equivalently (taking also (27) into account), there is s 0 ∈ {1, . . ., j} such that a ∼ j−s b whenever 1 s < s 0 and a > j−s 0 b, which means that (a, b) ∈ Inv j (ω(τ ) ). Case 2: Assume that i a > i b , i.e., a > j b.
Here, the couple (i b , i a ) is an inversion of τ if and only if one of the following conditions holds • (i a ) s = (i b ) s whenever 1 s m; or • there is s 0 ∈ {1, . . ., m} such that (i a ) s = (i b ) s whenever 1 s < s 0 and (i a ) s 0 < (i b ) s 0 .
In view of (27), this is equivalent to the single condition: • there is s 0 ∈ {1, . . ., j} such that a ∼ j−s b whenever 1 s < s 0 and a < j−s 0 b, which means that (a, b) ∈ Inv j (ω(τ ) ).
In both cases we have shown (26).The proof is complete.
τ ∈RTab(T ) x n inv (τ ) = 2 i p 1 j q d i,j−1 d i−1,j d i,j x , where d T = (d i,j ) is the p×q matrix of Theorem 20 (with d i,0 := 0).
Remark 30.(a) In view of Theorem 27 (and ( 5)), the generating function for inversion number on row-increasing tableaux χ T (x) := τ ∈RTab(T ) x n inv (τ ) is therefore realized as the Poincaré polynomial of the smooth, irreducible, projective variety Y T .The fact that χ T (x) is unimodal and palindromic (pointed out in [5, Corollaries 2.8-2.9])can then be viewed as a consequence of the Lefschetz theorem.(b) By Theorem 27, the maximal inversion number of an element τ ∈ RTab(T ) is dim Y T , and it is attained for a unique tableau τ 0 (see also [5,Corollary 2.7]).The equality dim H 2m (Y T , Q) = dim H 2(dim Y T −m) (Y T , Q) for all m = 0, . . ., dim Y T (which is due to the Lefschetz theorem, or to the fact that χ T (x) is palindromic) implies that there is an involution RTab(T ) → RTab(T ), τ → τ * such that n inv (τ * ) = dim Y T − n inv (τ ) for all τ ∈ RTab(T ).
In particular this involution must verify (τ 0 ) * = T .For arbitrary τ , we have no explicit formula for τ * .x n inv (τ ) = T ∈STab k (λ(u)) where m 0 := dim Fl k,u .Since the Spaltenstein variety Fl k,u is connected, we know that dim H 0 (Fl k,u , Q) = 1, hence m 0 is the maximal inversion number for the elements of RTab k (λ(u)) and it is attained for a unique tableau τ max .This tableau and its rectification T max := Rect(τ max ) are explicitly described in [4, §2.1].For this tableau we have dim Y Tmax = n inv (τ max ) = dim Fl k,u = dim Fl k,u,Tmax , which means that the affine bundle ϕ Tmax : Fl k,u,Tmax → Y Tmax must be an isomorphism.This implies that Fl k,u,Tmax is a projective (hence closed) subvariety of Fl k,u .Hence it is actually an irreducible component of Fl k,u which is smooth and isomorphic to the variety X(d Tmax , A) of Theorem 20.
In particular, every Spaltenstein variety contains at least one smooth irreducible component, which is isomorphic to a variety of the form X(d, A).
1 , and Theorem 9 retrieves the properties of the decomposition of this partial flag variety into Schubert cells.Specifically, the map w ∈ S n → ω(w) := ({w 1 , . . ., w d i,1 }) p i=1 ∈ W(d) yields a bijection between the set W(d) and the quotient S n /S d of the symmetric group by the parabolic subgroup S d := {w ∈ S n : w({1, . . ., d i,1 }) = {1, . . ., d i,1 } ∀i = 1, . . ., p}.In addition the inversion number n inv (ω(w)) coincides with the Coxeter length of the representative of minimal length of the coset wS d .(b) Next let us consider the special case where p = 2, which means that the dimension matrix d consists of two rows; let k := (d 1,1 . . .d 1,q ) be the entries in the first row; let := (d 2,1

( a ) 2 .
For every T ∈ STab k (λ(u)), the subset Fl k,u,T ⊂ Fl k,u is nonempty, locally closed, smooth, irreducible, of dimension equal to λ 1 (b) Therefore, the closures Fl k,u,T , for T ∈ STab k (λ(u)), are exactly the irreducible components of Fl k,u .Note that parts (b) and (c) of Proposition 13 are consequences of this result.Proposition 15 (a) can be proved by induction on n.The fact that the subsets Fl k,u,T are locally closed (and smooth) can also be shown as follows.The iterated kernels of u form an increasing sequence ker u ⊂ ker u 2 ⊂ . . .⊂ ker u λ 1 = C n , i.e., a partial flag.The stabilizer Q := {g ∈ GL n (C) : g(ker u j ) = ker u j ∀j} of this flag is a parabolic subgroup of GL n (C).By definition the number of boxes in the first j columns of the Young diagram λ(u| electronic journal of combinatorics 25(3) (2018), #P3.41
For a, b ∈ {1, . . ., d p,j }, we deduce the equivalences a < j b ⇔ i a < i b and a ∼ j b ⇔ i a = i b .Let 1 a < b d p,j and let us show the equivalence: (a, b) ∈ Inv j (ω(τ ) ) ⇔ (i a , i b ) or (i b , i a ) is an inversion pair for τ (26) (depending on whether i a < i b or i b < i a ); the desired formula n inv (τ ) = n inv (Ξ T (τ )) is clearly guaranteed once we show (26).Let ((i a ) 1 , . . ., (i a ) ) (resp.((i b ) 1 , . . ., (i b ) m ) be the list of entries directly to the right of i a (resp.i b ) in τ .Since i b is below i a in τ , we have m .As above, (i a ) s = min{i : a ∈ ω i,j−s }, (i b ) s = min{i : b ∈ ω i,j−s } for all s = 1, . . ., m and m is the minimal number such that b > d p,j−(m+1) (using the convention d p,0 = 0), so b > max{d p,j−(m+1) , a}, whence a < j−(m+1) b.