A transitivity result for ad-nilpotent ideals in type A
aa r X i v : . [ m a t h . R T ] J a n A TRANSITIVITY RESULT FOR AD-NILPOTENT IDEALS IN TYPE A MOLLY FENN AND ERIC SOMMERSA
BSTRACT . The paper considers subspaces of the strictly upper triangular matrices, which are stable under Liebracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalannumber of such subspaces. Each ad-nilpotent ideal I meets a unique largest nilpotent orbit O I in the Lie algebraof all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studiedby Mizuno and others, the equivalence classes are the ad-nilpotent ideals I such that O I = O for a fixed nilpotentorbit O . We include two applications of the result, one to the higher vanishing of cohomology groups of vectorbundles on the flag variety and another to the Kazhdan-Lusztig cells in the affine Weyl group of the symmetricgroup. Finally, some combinatorial results are discussed.
1. I
NTRODUCTION
Let G be a connected, simple algebraic group over C and B a Borel subgroup of G . Let g be the Liealgebra of G , b the Lie algebra of B , and n the nilradical of b . Fix a maximal torus T in B with Lie algebra h .Let W be the Weyl group of G relative to T .The paper is concerned with the subspaces of n which are stable under the adjoint action of B (or theadjoint action of b ). They are called ad-nilpotent ideals or B -stable ideals of n . We will refer to them simplyas ideals. Denote the set of all ideals by I d . Cellini and Papi [7] showed that the cardinality of I d is the W -Catalan number of G , namely | W | n Y i =1 ( h + d i ) , where d , . . . , d n are the fundamental degrees of W and h is the Coxeter number, which is the largest ofthe fundamental degrees; this reduces to the usual Catalan numbers in type A . These ideals play a role inmany structural results concerning nilpotent orbits in g and related objects in the representation theory of G ,including the partial order on nilpotent orbits under the closure relation [11], [20], [26], the representationsof the corresponding group over a finite field [16], the vanishing of odd cohomology of Springer fibers [8],cells in the affine Weyl group [24], Hessenberg varieties [5], [13], to name a few. They are also interestingfrom a purely combinatorial perspective since I d is in bijection with Dyck paths of length n + 2 in type A n ,as discussed in the last section.Let g.X := Ad ( g )( X ) denote the adjoint action of g ∈ G on X ∈ g and G.X the orbit of X under G . Anilpotent orbit in g refers to the orbit O of a nilpotent element. There are finitely many such orbits. Let I ∈ I d . Then I consists only of nilpotent elements. The G -saturation of I , denoted G.I := { g.X | g ∈ G, X ∈ I } ,is the closure of a single nilpotent orbit, denoted O I , and O I is also characterized as the unique nilpotentorbit such that O I ∩ I is dense in I . We call O I the associated (nilpotent) orbit of I . Conversely, every nilpotentorbit arises as O I for some I by the Jacobson-Morozov theorem. See, for example, [26] for these results.The partial order on the set all nilpotent orbits is defined by containment of closures (that is, O ≤ O ifand only if O ⊂ O ) . Clearly, if I ⊂ J , then O I ≤ O J . Gerstenhaber computed the partial order for classicalgroups [11]. There, he introduced I d , which were called triangular subalgebras, and described an algorithmfor finding O I in type A n . We review Gerstenhaber’s algorithm and give a new proof in Section 6. In [20]Mizuno described the partial order on nilpotent orbits in the exceptional Lie algebras and computed thecomponent groups of nilpotent elements by studying many operations on I d that preserve the associatednilpotent orbit. This paper is concerned with the simplest operation (see Definition 2.1), which we call the basic move . This operation is also important in studying cohomology of vector bundles on G/B [4]. Ourmain result (see Theorem 2.2) is that in type A n this operation is transitive on the set of all I ∈ I d withthe same associated orbit. A transitivity result of this kind is proved in [8] for some of the ideals in theexceptional groups when O I is distinguished. We give two applications of Theorem 2.2 in Section 5 anddescribe some combinatorially results in Section 6. In other types, additional operations on I d were studied in the PhD thesis of the first author [10], where itis conjectured that an analogous transitivity result holds when these additional operations are allowed. Thishas been checked in the exceptional groups by computer, but is still open for other classical types. Indeed,if one knows such transitivity results, then the partial order on nilpotent orbits reduces to a combinatorialproblem involving only I d . Namely, consider equivalence classes [ I ] on I d defined by the condition O I = O . The partial order on I d by inclusion of ideals induces a partial order on equivalence classes in I d . Then, [ I ] ≤ [ J ] if and only if O I ≤ O J . This was proved uniformly in [26]. Of course, the idea is exactly the oneused by Gerstenhaber and Mizuno to explicitly compute the partial order.2. B ASIC MOVE
Let Φ ⊂ h ∗ be the roots of G on g relative to T . Let Φ + and Π be the positive roots and simple rootsdetermined by B . For β ∈ Φ , let g β be the β weight space in g .Since ideals are T -stable, any I ∈ I d decomposes as sum of root spaces. Define I ⊂ Φ + by the equation I = M β ∈I g β . (1)Let α ≺ β denote the usual partial order on roots; that is, α ≺ β if and only if β − α is a sum of positiveroots. Then I has the property that if α ∈ I and α ≺ β , then β ∈ I . In turn, any I ⊂ Φ + with this propertydetermines a unique I ∈ I d via equation (1). Such an I is an upper order ideal in the poset Φ + and we referrefer to them as root ideals or just ideals and go back and forth between I and its root ideal I . The set ofminimal roots in I form an antichain (no two distinct elements are comparable) and each I is determinedby its antichain. Let I min denote the minimal roots of I .For α ∈ Π , let P α be the parabolic subgroup of G containing B attached to α and let s α ∈ W be the simplereflection attached to α . Then P α /B is a projective line. Given distinct I, J ∈ I d , suppose there exists α ∈ Π such that P α .J = I . Then necessarily J ⊂ I and (1) dim I = dim J + 1 since P α /B is dimension one; (2) I is stable under P α ; and (3) O I = O J since O I meets J and O J ≤ O I . For such a pair of ideals to exist,we must have I = J ⊕ g β with β a minimal root of the associated root ideal I , and s α ( I ) = I . Hence, s α ( β ) = β + kα ∈ I and thus k < since k = 0 contradicts P α .J = J and k > contradicts that β ∈ I min .Thus h β, α ∨ i ∈ {− , − , − } . For some applications, it is useful to insist that this value is − (of course, thisis always true in type A and other simply-laced types). Definition 2.1.
Let I ∈ I d and I the associated root ideal. Let α ∈ Π and β ∈ I min with h β, α ∨ i = − . Then J := I \ { β } is a root ideal (with associated ideal J ∈ I d ). If s α ( I ) = I , then O J = O I and we say that I and J (or I and J ) are related by the basic move and write I ∼ J (respectively, I ∼ J ).We can now state the main result.
Theorem 2.2.
Let
I, J ∈ I d . In type A n if O I = O J , then I and J are related by a sequence of basic moves. That is,the equivalence relation on I d induced by the transitive action of basic moves coincides with the equivalence relationon I defined by having the same associated nilpotent orbit.
3. P
RELIMINARIES
From now on, let G be the general linear group GL n +1 ( C ) and B the upper triangular matrices and T the diagonal matrices in G . Let t ij be the matrix with in the ( i, j ) spot and zero elsewhere. Take thestandard basis { e i } of h ∗ so that the simple roots are α i = e i − e i +1 for ≤ i ≤ n . Each positive roothas the form α i + α i +1 + · · · + α j = e i − e j +1 for ≤ i < j ≤ n , corresponding to the weight space in g = M n +1 ( C ) spanned by t ij , and we denote this root by [ i, j ] . The usual partial order on the positive rootsis expressed as [ i, j ] (cid:22) [ i ′ , j ′ ] if and only if i ′ ≤ i and j ≤ j ′ . Since a root ideal I is determined by theminimal roots in I under the partial order, we can specify an ideal by its collection of minimal roots andwrite I min = { [ a k , b k ] } for the minimal roots. If [ a, b ] ∈ I min we say a is a left endpoint of I and b is rightendpoint of I . If s α j ( I ) = I , we say I is j -stable. This translates to saying j is neither a right endpoint, nora left endpoint, of I .Let P be a standard parabolic subgroup of G , i.e., a closed subgroup containing B . Let n P denote thenilradical of the Lie algebra of P . Then n P ∈ I d . We will also denote by n P the associated root ideal. All ofthe minimal roots of n P are simple roots, and conversely if I min are all simple, then I is equal to some n P . TRANSITIVITY RESULT FOR AD-NILPOTENT IDEALS IN TYPE A We can designate P := P j ,...,j l in terms of the minimal roots α j , . . . , α j l of n P where ≤ j < · · · < j l ≤ n .The sequence ( j , . . . , j l ) gives rise to a composition of n + 1 with l + 1 parts:(2) ( c , . . . , c l , c l +1 ) := ( j , j − j , . . . , j l − j l − , ( n + 1) − j l ) . The standard Levi subgroup of P j ,...,j l is isomorphic to Q i GL c i , embedded as block diagonal matrices in G . Denote by µ P the partition of n + 1 obtained by arranging the composition in (2) in weakly descendingorder.The proof of Theorem 2.2 proceeds by showing that every I ∈ I d is equivalent to some n P . Then it isshown that n P ∼ n P ′ if and only if µ P = µ P ′ .We start with some lemmas. Let [ a, b ] ∈ I min . Omitting the root [ a, b ] from I gives a new ideal J . Let A = I min \{ [ a, b ] } . Then J min = A ∪ S where S ⊂ { [ a − , b ] , [ a, b + 1] } specified by [ a − , b ] ∈ S if a − isnot nor a left endpoint of I min and [ a, b + 1] ∈ S if b + 1 is not n + 1 nor a right endpoint of I min . We nowdescribe the basic move from Definition 2.1 in type A n . Definition 3.1 (Basic Move in type A n ) . Let I be a root ideal and [ a, b ] ∈ I min . If I is ( a − -stable or ( b +1) -stable. Then I and J := I\{ [ a, b ] } are equivalent under the basic move (2.1) and we write I ∼ J or I min ∼ J min .We record a few lemmas that follow quickly from the basic move. Lemma 3.2.
Let I be an ideal with I min = A ∪ { [ a, j ] , [ b, j + 1] } for a < b ≤ j . Then [ b, j ] is not comparable toany root in A so there is an ideal J with J min = A ∪ { [ b, j ] } . Assume the intervals in A contain no endpoints j with a < j < b . Then I ∼ J .Proof.
The hypothesis on A means that J is ( b − -stable, so J min ∼ A ∪ { [ b − , j ] , [ b, j + 1] } . Set A ′ = A ∪ { [ b, j + 1] } . Then using the basic move multiple times gives A ∪ { [ b − , j ] , [ b, j + 1] } ∼ A ′ ∪ { [ b − , j ] } ∼ · · · ∼ A ′ ∪ { [ a, j ] } since A ′ does not contain b − , b − , . . . , a + 1 . (cid:3) A similar argument gives
Lemma 3.3.
Let
I ∈ I d with I min = A ∪ { [ a, n ] } . Let J have J min = A ∪ { [ b, n ] } . Assume the intervals in A contain no endpoints j with a < j < b . Then I ∼ J . The result also holds if [ a, n ] is replaced by [1 , b ] and [ b, n ] by [1 , a ] in the definitions of I and J .Example . In A , we have { [1] } ∼ { [1 , } ∼ { [1 , } and { [3] } ∼ { [2 , } ∼ { [1 , } by Lemma 3.3 with A = ∅ . And { [2] } ∼ { [1 , , [2 , } by Lemma 3.2 with a = 1 , b = 2 , j = 2 and A = ∅ . Finally, { [1] , [2] } ∼{ [1] , [2 , } ∼ { [1] , [3] } ∼ { [1 , , [3] } ∼ { [2] , [3] } via basic moves. Each of ∅ and { [1] , [2] , [3] } are their ownequivalence class and this accounts for the classes among the elements of I d . Also each class containsat least one n P . 4. P ROOF OF T HEOREM S ⊂ Φ + a right staircase if there exist positive integers c ≤ d and a i ’s such that(3) S = { [ a i , i ] | c ≤ i ≤ d } and either d = n or c ≤ a d . Note that a i < a j for c ≤ i < j ≤ d . Examples of such S in A are { [2 , , [3 , , [6 , } and { [1 , , [3 , , [4 , } . Lemma 4.1. If I min contains a right staircase, then no minimal root of I can have left endpoint j with a c ≤ j ≤ c unless j = a i for some i .Proof. If [ j, k ] were a minimal root with a c ≤ j ≤ c , then k > c ; otherwise [ j, k ] ≺ [ a c , c ] . Furthermore k > d since only one minimal root can have a given right endpoint and the values i with c ≤ i ≤ d are spoken for.But if k > d , then d = n and we must have c ≤ a d . But then [ a d , d ] ≺ [ j, k ] since j ≤ c ≤ a d and k > d , whichis a contradiction. (cid:3) Lemma 4.2.
Every ideal is equivalent to an ideal containing a right staircase.
MOLLY FENN AND ERIC SOMMERS
Proof.
Let [ a, b ] be the interval of I with largest b . If b < n , Then I is b + 1 -stable, so apply the basic moveand omit [ a, b ] to obtain an equivalent ideal with interval [ a, b + 1] . Repeating this process, we see that theequivalence class of I contains an ideal with interval [ a, n ] , which qualifies as a right staircase. (cid:3) If I min contains a right staircase S and furthermore no minimal root of I has right endpoint j with a c ≤ j ≤ c − , then we say S is a pure right staircase in I min . Lemma 4.3.
Suppose I min contains the pure right staircase S := { [ a i , i ] | c ≤ i ≤ d } . Then I is equivalent to an ideal containing the simple root α c = [ c, c ] = [ c ] .Proof. We can assume a c < c , otherwise [ c ] ∈ S and there is nothing to show.Suppose a d < c . We will show that I is equivalent to an ideal containing the same staircase except that c ≤ a d . Since a d < c , then d = n from the definition of a right staircase. For any j with a d < j < c , j cannotbe a right endpoint by the pure hypothesis and the fact that a c ≤ a d ; nor can j be a left endpoint by Lemma4.1. This means [ a d , n ] can be replaced in I by [ c, n ] to yield an equivalent ideal by Lemma 3.3, completingthis part of the argument.Next, we show that if c < a d , then I is equivalent to an ideal containing a pure staircase with a d = c .Let j be the largest number with the property that a j < c . Such a j exists since we assumed a c < c . Then a c ≤ a j ≤ c − . By the pure assumption a j does not occur as the right endpoint of a minimal root of I .Since c < a d , j = d and thus [ a j +1 , j + 1] is in the staircase. Now we can apply (the second part of the proofof) Lemma 3.2 to replace [ a j , j ] with [ c, j ] since k is neither a right nor left endpoint in I for a j < k < c . Thisshows that I is equivalent to an ideal containing a pure staircase with a d = c since we can just forget all thelater roots in the original staircase.If c = d , we are already done. If not, we can apply Lemma 3.2 to [ a d − , d − and [ a d = c, d ] , replacingthose with [ c, d − . The proof follows by induction on the difference d − c and we arrive at the interval [ c ] ,as desired. (cid:3) Example . The steps in the lemma for { [2 , , [3 , , [6 , } are with c = 5 : { [2 , , [3 , , [6 , } ∼ { [2 , , [5 , , [6 , } and then { [2 , , [5 , , [6 , } ∼ { [5] , [6 , } . Corollary 4.5.
For any k ≥ n − , the root ideal I := { [ j, j + k ] | ≤ j ≤ n − k } is equivalent to { [ k + 1] } and also to { [ n − k ] } .Proof. This is a pure staircase with c = k + 1 with a d = n − k . Since n − k ≤ k + 1 , we have a d ≤ c , so thefirst part of the previous proof implies that I is equivalent to a pure staircase with a d = c . Then the lastparagraph implies I ∼ { [ c ] } . A symmetric argument as in the lemma but moving right endpoints insteadof left gives I ∼ { [ n − k ] } . (cid:3) Proposition 4.6.
Every ideal I is equivalent to an ideal with a minimal root which is a simple root. Hence, everyideal is equivalent to some n P .Proof. By Lemma 4.2, we can assume that I contains a right staircase. Next, we prove that if I contains aright staircase S = { [ a i , i ] | c ≤ i ≤ d } that is not pure, then I is equivalent to an ideal with a right staircasewith smaller value of c . Then the proposition will follow: either we encounter a right staricase that is pureand we can apply Lemma 4.3, or we eventually arrive at a staircase with c = 1 , which means a c = c = 1 ;that is, [1] ∈ I min .Therefore, assume that S is not pure. Then there exists a largest m < c with a c ≤ m so that [ l, m ] ∈ I min .Define L = { j | m < a j < c } . If L is not empty, let j be its smallest element. Then we can apply theargument from Lemma 4.3 to the right staircase S ′ = { [ a i , i ] | j ≤ i ≤ d } to replace it with [ c, j ] arrivingat an equivalent ideal containing [ a m , m ] and no endpoints strictly between m and c . This statement is alsotrivially true if L is empty. TRANSITIVITY RESULT FOR AD-NILPOTENT IDEALS IN TYPE A Now, if m = c − , we are done; otherwise I is stable under m + 1 . And so we can replace [ a m , m ] byan equivalent ideal containing [ a m , m + 1] which has no endpoints strictly between m + 1 and c . It followsby induction on c − m that I is equivalent to an ideal with a right staircase that begins with [ a m , c − andends with either [ c, j ] ( L not empty]) or [ a d , d ] ( L empty). This completes the proof of the first statement.Now, given I ∈ I , it is equivalent to an ideal with minimal simple root α i for some i . The other minimalroots for I must all come from one of the two irreducible root subsystems generated by the remainingsimple roots, which are of type A i − and A n − i . By induction on n we can assume the theorem in types A i − and A n − i , and hence the ideal I is equivalent to one whose minimal roots are all simple. (cid:3) Lemma 4.7.
For standard parabolic
P, P ′ , we have n P ∼ n ′ P if and only if µ P = µ ′ P .Proof. Let the minimal roots of n P be [ j ] , . . . , [ j l ] . Set j = 0 and j l +1 = n + 1 . Let ≤ b ≤ l . ThenCorollary 4.5 applies to each subsystem with simple roots α i for j b − + 1 ≤ i ≤ j b +1 − . Namely, It impliesthat n P is equivalent to n P ′ with the same minimal roots except that [ j b ] is replaced with [ j b − + j b +1 − j b ] .The composition for n ′ P is the same as the one for n P except that the j b +1 − j b and j b − j b − exchange(adjacent) positions. Thus this action acts like the simple transposition ( b, b + 1) on the l + 1 elements ofthe composition, generating the symmetric S l +1 . This shows all rearrangements of the composition yieldequivalent ideals and therefore that if µ P = µ ′ P , then n P ∼ n P ′ . Together with the Proposition 4.6, we knowthat the number of equivalence classes is at most n + 1 .On the other hand, the number of nilpotent orbits equals the number of partitions of n + 1 and everyorbit arises as O I for some I ∈ I d (see the introduction). Hence the number of equivalence classes is at leastthe number of partitions of n + 1 . We conclude the number of classes is exactly the number of partitionsof n + 1 . Hence the converse statement is also true: distinct partitions correspond to distinct equivalenceclasses. (We will recall in Section 6.1 that the Jordan type of O n P is the dual partition of µ P ). (cid:3) The proof of Theorem 2.2 is now complete by the proof of Lemma 4.7: the equivalence classes under thebasic move coincide with the equivalence classes under associated nilpotent orbit.5. T
WO APPLICATIONS
Vanishing cohomology.
For a rational representation V of B , denote by H ∗ ( G/B, V ) , the cohomologyof the sheaf of sections of the vector bundle G × B V over G/B . Let V ∗ denote the linear dual of V and S j V ∗ the j -th symmetric power of V ∗ , which are all B -modules.For each parabolic subgroup P containing B , it is known in all Lie types that H i ( G/B, S j n ∗ P ) = 0 for all j ≥ and i > (see [15, Chapter 8]). By Proposition 4.3 in [1], if I and J are related by the basic move, then(4) H i ( G/B, S j I ∗ ) = H i ( G/B, S j J ∗ ) for all i, j ≥ . Hence a corollary of Theorem 2.2 is the following.
Corollary 5.1.
In type A n , every B -stable ideal I of n satisfies H i ( G/B, S j I ∗ ) = 0 for all j ≥ and i > . We expect this to hold in all types and it has been checked for most ideals in the exceptional groups [10].The vanishing result is already known in all types for the ideal g ≥ attached to each orbit via the Jacobson-Morozov theorem [14], [22], so the corollary gives a new proof for g ≥ in type A . The vanishing also impliesformulas for the G -module of graded functions on O I . Let I c := Φ + \I be the lower order ideal. Then forany I in type A , the ungraded functions on O I satisfy R ( O I ) = Ind GT Y α ∈I c ( e − e α ) ! in the notation of Corollary 3.2 in [19]. A graded formula is obtained by the Kostant multiplicity formulausing Lusztig’s q-analog of weight multiplicity, but where the positive roots allowed are drawn only from I (see [15, Chapter 8]). MOLLY FENN AND ERIC SOMMERS
Cells in the affine Weyl group.
Shi [24] computed the left cells in the affine Weyl group of type A n .The left cells, viewed geometrically, are unions of regions of the Shi arrangement. In the dominant chamber,the regions of the Shi arrangement are indexed by positive sign types and these are in bijection with I d . For I ∈ I d , let R I be the associated region. Then Shi has two results: (1) for I ∈ I d , the region R I lies in a leftcell; (2) if I and J are related by a certain equivalence relation [24, p. 103], then they lie in the same left cell.Assume Shi’s first result above. Then Fang showed [9, Theorem 4.3] that if I and J are related by thebasic move, then R I and R J lie in the same left cell. Hence Theorem 2.2 implies Corollary 5.2.
In type A n , the regions R I and R J lie in the same left cell whenever O I = O J . Hence, Shi’sequivalence relation for the positive sign types coincides with the one generated by the basic move. In fact, the partition λ I is the same as the partition attached by Lusztig to the two-sided cell that contains R I [18]. So the corollary is a variant of the intepretation of the partition in terms of nilpotent orbits for theloop group by Lawton [17]. Fang’s result is based on Lusztig’s star action, which is analogous to Knuth’srelation on S n . 6. C OMBINATORICS
Gerstenhaber’s algorithm.
For a partition µ of n + 1 , let O µ denote the conjugacy class of nilpotentmatrices with Jordan blocks of sizes equal to the parts of µ . If x ∈ O µ , we write λ ( x ) for µ . Define λ ( O I ) := λ ( x ) for any x ∈ O I . Recall that O µ ≥ O ν if and only if µ ≥ ν in the dominance order on partitions.Given I ∈ I d , there is an algorithm to compute λ ( O I ) due to Gerstenhaber [11, p. 535]. The algorithmproduces k disjoint, ordered subsets S , . . . , S k of { , , . . . , n + 1 } , called the characteristic sequences of I .Each sequence is defined inductively. The first element of S is . Then if i ∈ S , its successor j is defined tobe the smallest integer such that t ij ∈ I , or equivalently, e i − e j ∈ I . If no j exists, the sequence terminates.Once S , . . . , S r are defined, for any matrix we can cross out the rows and columns indexed by the elementsin ∪ ri =1 S i . This defines a map p : gl n +1 → gl s , where s = n + 1 − ∪ ri =1 S i , which takes ideals to ideals. Then S r +1 is defined as S was, but keeping the original labelling of the rows and columns. The ideal propertyof I ensures that | S i | ≥ | S i +1 | and Gerstenhaber’s partition of n + 1 is defined as λ I := ( | S | , | S ] , . . . ) . Foreach characteristic sequence S j = { i , . . . , i r } , let x j = t i ,i + t i ,i + · · · + t i r − ,i r . Then the nilpotent matrix(5) x := k X j =1 x j ∈ I, and clearly λ I = λ ( x ) by construction. Gerstenhaber proved [11, Theorem 1] that x ∈ O I , showing that λ ( O I ) coincides with λ I . We can give a new proof of this result using Theorem 2.2. Proposition 6.1.
The partition λ I is unchanged under basic moves.Proof. Suppose I and J are related by the basic move, where the weight space of e i − e j is dropped from I to yield J . We need to show λ I = λ J . This is clear if e i − e j is not selected in any step of the algorithmproducing λ I . If it is selected, i.e., j follows i in some characteristic sequence for I , then there are two cases:when I is ( i − -stable and when I is j -stable.Case 1: I is stable under the action of s i − = ( i − , i ) . In particular i ≥ . We will show that actually j cannot follow i in any characteristic sequence. First, e i − − e b
6∈ I for any i − < b < j . For b = i , this isclear since I is s i − stable. For i < b < j , we would get s i − ( e i − − e b e )) = e i − e b ∈ I . But e i − e j is a minimalroot of I , so this cannot happen. Second, i − cannot be selected in an earlier characteristic sequence than i for I . Otherwise, j would be available to follow i − at the earlier step since e i − − e j ∈ I ; hence j willbe chosen since it is the minimal such value by the first step. This contradicts that j follows i in a sequence.Next, i cannot begin a characteristic sequence since i − would be available to start the sequence by thesecond step, and thus it must be chosen as the first element, ahead of i . Finally, if i is part of a sequence thatincludes a, i, j , then a = i − as before (by s i − stability of I ). While if a < i − , then e a − e i ∈ I implies s i − ( e a − e i ) = e a − e i − ∈ I . Hence, i − would have followed a since i − was available to be chosen(intead of i ). Thus, stability of I under s i and e i − e j minimal in I means j cannot follow i in any sequence.We conclude that the characteristic sequences for I and J are identical.Case 2: I is stable under s j = ( j, j + 1) . Let j follow i in the r -th sequence for I . Suppose j + 1 appearsin an earlier sequence for I . First, it cannot start that sequence since i < j + 1 and i being available means i or a smaller number would be selected to start the sequence. TRANSITIVITY RESULT FOR AD-NILPOTENT IDEALS IN TYPE A Next, if j + 1 follows some a , then e a − e j +1 ∈ I and then also e a − e j ∈ I by s j -stability ( a = j since j occurs in a later sequence). But e i − e j is minimal in I , so a < i . But then j would have followed a in thisearlier sequence, instead of j + 1 , a contradiction. We conclude that j + 1 must appear in a later sequencefor I (it cannot occur in the same sequence as j since e j − e j +1
6∈ I by s j -stability).Now, since e i − e j is omitted from I and j + 1 is still unchosen at the r -th step, the r -th sequence for J must have j + 1 following i (note that e i − e j +1 ∈ I and hence also in J ). If some number, say b , follows j in the algorithm for I , then e j − e b ∈ I . And then s j -stability means e j +1 − e b is in both I and J . Hence, s follows j + 1 in the r -th sequence for J . This shows that the r -th part for λ I and λ J are the same. Also theearlier sequences are identical.The remainder of λ I and λ J are determined by ideals in gl n +1 − s under p . We claim they are the sameideal and that the labeling is the same except that the j label for p ( J ) becomes the j +1 label for p ( I ) . Indeed, e a − e b ∈ p ( J ) means e a − e b ∈ J and a and b are not equal to j + 1 . Hence, the indices in s j ( e a − e b ) avoid j .Since I is s j -stable, that means s j ( e a − e b ) ∈ I and then also in p ( I ) . The converse is similar, concluding theproof that λ I = λ J if I ∼ J . (cid:3) Corollary 6.2 (Theorem 1 in [11]) . We have λ ( O I ) = λ I .Proof. By Proposition 2.2 and Lemma 4.7 and the previous proposition, this reduces to computing both λ I and λ ( O I ) for I := n P where P = P j ,j ,...,j l is such that the composition c i in (2) is already a partition. Let µ := ( j , j − j , . . . , j l − j l − , n + 1 − j l ) be this partition.Recall the dual partition µ ∗ of µ is defined by µ ∗ j = { i | µ i ≥ j } . So µ ∗ = l + 1 . It is easy to see that λ I = µ ∗ (as shown in [11, Proposition 14]). Namtely, set j = 0 . Then the first characteristic sequence is ( j + 1 , j + 1 , j + 1 , . . . , j l + 1) , the second is ( j + 2 , j + 2 , . . . , j µ ∗ − + 2) , and so on. The i -th sequencehas length µ ∗ i , so that λ I = µ ∗ .Next, the matrix x ∈ I in (5) has λ ( x ) = λ I by construction. Hence x ∈ O µ ∗ . The dimension of O µ ∗ is ( n + 1) − P µ i [6]. Then by Richardson, dim( O I ) = 2 dim( n P ) [6] and the latter equals dim GL n +1 − dim( L ) where L ≃ Q GL µ i . Hence, dim( O I ) = dim( O µ ∗ ) , so it must be that O I = O µ ∗ . Therefore, λ ( O I ) = µ ∗ = λ I . (cid:3) Two coarser equivalence relations.
First, we recall the bijections among I d , lattice paths, Dyck paths,and ballot sequences. Let { b i } ni =1 be a sequence consisting of zeros and ones. The height h j := j X i =1 ( − b i +1 of the sequence at index j is the number of ’s minus the number of ’s in the subsequence { b i } ji =1 . A binarysequence { b i } ni =1 is called a ballot sequence of length n if there are n zeros and n ones and h j ≥ for all j .The maximum height of the ballot sequence is the largest value of h j .The ballot sequences of length n give rise to a lattice path in the plane as follows: starting at (0 , n ) thepath moves a unit step east at time i if b i = 1 and a unit step south if b i = 0 . Since there are n ones and n zeros, the path terminates at ( n, . Since h j ≥ for all j , the path stays at or above the line joining (0 , n ) and ( n, . This is a bijection between ballot sequences and such lattice paths. Such lattice paths are then inbijection with the ideals in gl n . The corresponding ideal I is the one whose border is this lattice path wherethe lower left corner of a matrix is at (0 , ; namely, where t ij ∈ I if and only if the point ( n − i, j ) lies onor northeast of the lattice path. Finally, these lattice paths are clearly in bijection with the Dyck paths in theplane, which start at (0 , and move by (1 , is b i = 1 and by (1 , − if b i = 0 and stay above the x -axis.Then the notion of maximum height is just the y -coordinate of the largest peak in the Dyck path.The first characteristic sequence ( i = 1 , i , . . . , i r ) of I was subsequently re-introduced in the literature[2] and then adapted to Dyck paths [12], where it determines the bounce path . Visually, this is the lattice paththat traces the lower boundary of the ideal I := n P where P = P i − ,...,i r − . The value ( λ I ) − r − iscalled the bounce count or bounce number of the path since this is the number of times the path touches thediagonal (strictly inside the matrix); namely, it touches at the points ( i − , i − , . . . , ( i r − , i r − .Next, we break the basic move down into two coarser moves. Definition 6.3.
Let [ i, j −
1] = e i − e j be a minimal root of I .If either MOLLY FENN AND ERIC SOMMERS (1) i ≥ and e a − e i is not a minimal root for all a < i , or(2) j ≤ n and e j − e b is not a minimal root for all b > j ,then we say I and I\{ e i − e j } are equivalent under the inner move .If either(1) i ≥ and e i − − e a is not a minimal root for all a > i − , or(2) j ≤ n and e b − e j +1 is not a minimal root for all b < j + 1 ,then we say I and I\{ e i − e j } are equivalent under the outer move . Proposition 6.4. If I and J are equivalent under the inner move, then ( λ I ) = ( λ J ) . In the language of Dyckpaths, the bounce count is invariant under the inner move.Proof. Assume we are omitting e i − e j from I by the inner move. We need only be concerned if e i − e j isselected in the constructing the first characteristic sequence for I .If the first condition of the inner move applies, then i ≥ so i does not begin the sequence. If ( a, i, j ) appears in the first sequence, then e a − e i ∈ I . But this root cannot be minimal, so let e b − e c be a minimalroot below it in the partial order, i.e., b ≥ a and c ≤ i . If c < i , then e a − e i − ∈ I , so i − would be chosento follow a , instead of i , in the algorithm since i − < i . Hence, c = i , which contradicts the condition onminimal roots. We conclude j cannot follow i in the first characteristic sequence.If the second condition of the inner move applies, the proof is same as part of Case 2 in the Proposition6.1. First, j + 1 cannot be selected in the first sequence; otherwise, e j − e j +1 ∈ I , hence minimal, which is notallowed. Thus, j + 1 follow i in the first sequence of J . Finally, if b follows j in the first sequence, then asin the proof of the proposition, b follows j + 1 in the first sequence of J since the argument only used that e j − e b
6∈ I min . (cid:3) Proposition 6.5.
The number of parts of λ I is the maximum height of the associated ballot sequence (or Dyck path).Proof. First, we show that the outer move preserves the maximum height of a ballot sequence. In thelanguage of ballot sequences, the minimal roots of I correspond to spots with b r = 0 and b r +1 = 1 . Set j = P k ≤ r b k , the number of ’s up to and including the in this subsequence. Set i = r − j , the numberof ’s up to and including this subsequence. Then e i − e j is a minimal root of I . The first condition ofthe outer move means that any subsequence extends on its left to , that is, b r − = 0 . The secondcondition of the outer move says that the subsequence extends on its right to , that is, b r +2 = 1 . Now,dropping this minimal root from the ideal changes the ballot sequence by replacing with . If either thefirst or the second condition of the outer move hold, so that changes to or changes to , themaximum height of the sequence is clearly unchanged.Next, I = n P as in Corollary 6.2 has maximum height equal to µ = ( λ I ) ∗ , i.e., the number of partsof λ I . By Theorem 2.2 any ideal is equivalent to some n P by a sequence of basic moves, hence the resultfollows since outer moves (hence basic moves) preserve maximum height and basic moves preserve λ I byProposition 6.1. (cid:3) Remark . Both bounce count and maximum height have an algebraic interpretation. Since λ ( O I ) = λ I by Corollary 6.2, it is clear that ( λ I ) is the smallest positive integer k such A k = 0 for all A ∈ I . Moreover,it is pointed out in [11, p. 536] that ( λ I ) is the index of nilpotence of I as an associative algebra , while in[2] it is shown that ( λ I ) is the index of nilpotence of I as a Lie algebra (actually, the latter use the class ofnilpotence to refer to the number of brackets needed to get to zero, which equals ( λ I ) − , i.e., the bouncecount). We will say that I has index k .On the other hand, the number of parts of λ I is the smallest k such that A has rank at most n − k for all A ∈ I . We say that I has corank k .In [2, Section 5] a bijection ι : I d → I d was introduced, where it was shown that ι sends an ideal ofbounce count k − to one of maximum height k . This has a nice interpretation in terms of Proposition 6.5.Namely, for each ≤ k ≤ n + 1 ,(6) ι ( { I ∈ I d | ( λ I ) = k } ) = { I ∈ I d | ( λ ∗ I ) = k } , so that ι sends an ideal of index k to one of corank k . The map ι also shows up as the inverse to the sweepmap on Dyck paths [12]. TRANSITIVITY RESULT FOR AD-NILPOTENT IDEALS IN TYPE A Remark . We wonder whether the inner moves generates the equivalence class of ideals of index k (i.e.,Dyck paths with bounce count k − ), and whether the outer move generates the equivalence class of idealsof corank k (i.e., Dyck paths with maximum height k ).6.3. Unit Interval Orders.
Another incarnation of I d in type A n is via unit interval orders. These are certainpartial orders on the set { , , . . . , n + 1 } . Given I ∈ I , the corresponding unit interval order P I is definedby i ≺ j if and only if t ij ∈ I .Consider a set partition of P I into a union of k disjoint chains C ∪ · · · ∪ C k with C i ≥ C i +1 . Let µ be the partition with µ i = C i . Then we can define a nilpotent element x ∈ I as in (5) and x satisfies λ ( x ) = µ . Hence λ I ≥ µ since λ I = λ ( x I ) ≥ λ ( x ) by Corollary 6.2. On the other hand, ∪ ki =1 S i , where S i areas in §6.1, is a disjoint union of k chains in P I . Therefore P ki =1 ( λ I ) i is the maximal cardinality of any suchdecomposition of P I . Thus (see [3]) Proposition 6.8. λ I is the Greene-Kleitman partition attached to P I . There is another perspective from the point of view of the indifference graph of P I (see [5]). This is theundirected graph G I on { , , . . . , n + 1 } where there is an edge between i and j if and only if i and j are notcomparable in P I . Clearly any decomposition for P I in k disjoint chains corresponds to a decomposition of G I into a disjoint union of k independent sets, and vice versa. The vertices in each independent set can becolored the same color. This means any coloring of G I gives rise to a chain decomposition, and vice versa.In particular, the largest part of λ I gains another interpretation as the independence number of G I andthe number of parts of λ I is the chromatic number of G I . Tim Chow (private communication) proposed acoloring algorithm for G I that is equivalent to the one in §6.1 and showed, without reference to nilpotentorbits, that λ I dominates the partition associated to any coloring of G I .Finally, we point out that the first numbers of the S i in §6.1 form an antichain in P I , which implies thenumber of parts of λ I is the size of any maximal antichain in P I and therefore equals the clique number of G I (which then also equals the chromatic number).6.4. Enumeration.
Several results related to enumerating the ideals of index k , hence also those of corank k , are given in [2]. It would be nice to be able to find a formula for the cardinality N λ of the equivalenceclass attached to O λ . We the list the values of N λ for low ranks in Table 1. For n ≤ , we have N λ = N λ ∗ ,but the numbers start to diverge for larger n . Still, we know from (6) that for all k (7) X λ : λ = k N λ = X λ : λ = k N λ ∗ , so there may be some further connection between N λ and N λ ∗ . From the last section, Equation (7) can alsobe phrased in terms of indifference graphs on unit interval orders, using independence number and cliquenumber in place of bounce count and maximum height.There is a second partition attached to an ideal coming from the minimal roots in I . Namely, if we define e I := X β ∈I min e β , then we can also associate to I the partition λ ( e I ) , which we call the Kreweras partition of I . The element e I can be defined for any Lie type. Since I min are the simple roots of the parabolic subsystem of Φ they span[25, Theorem 1], the element e I is always regular in a Levi subalgebra (outside of type A , the elements in O I are not always regular in a Levi subalgebra). Since e I ∈ I , in type A n it follows that(8) λ I ≥ λ ( e I ) . Unlike N λ , there are closed formulas for K λ := { I ∈ I d | λ ( e I ) = λ } . In fact, the formulas exist in allLie types, where they involve the exponents of a hyperplane arrangement attached to the parabolic rootsystem determined by I min [25, Proposition 6.6]. In type A n , they coincide with the Kreweras numbers,which are given in terms of multinomials by K λ = 1 n + 2 (cid:18) n + 2 a ( λ ) , a ( λ ) , . . . , a n +1 ( λ ) , n + 2 − ℓ (cid:19) , where a j ( λ ) denotes the multiplicity of the number j among the parts of λ and ℓ is the total number ofparts. A A A A λ N λ λ N λ λ N λ λ N λ [7] [6 , [5 , [5 , ] [4 , [6] [4 , , [5 , [4 , ] [4 , [3 , [4 , ] [3 , ] [3 ] [5] [3 , , ] [3 , , [4 , [3 , ] [3 , ] [3 , [4] [2 , [2 ] [3 , ] [3 , [2 , ] [2 , ] [2 , [2 ] [2 , ] [2 , ] [2 , ] [2 , ] [1 ] [1 ] [1 ] [1 ] ABLE
1. Values of N λ . ( λ I ) , m I
10 9 8 7 6 5 4 3 2 1 0 ( λ I ) ∗ , m I ABLE
2. Joint valley-bounce count (or valley-maximum height) statistics for A Let m I denote the number of minimal roots of I . Then m I equals the rank of e I , which is ( n + 1) − λ ( e I ) ∗ .In terms of Dyck paths, m I is the number of valleys of the Dyck path, while m I + 1 is the number ofpeaks. Then { I ∈ I d | m I = k } give the Narayana numbers ([23, Proposition 4.1]). We make the followingobservation based on analyzing the construction of ι in [2]. Proposition 6.9.
With respect to ι , we have (9) m I = n − m ι ( I ) for I ∈ I d . Putting (9) together with (6) we get (10) ι ( { I ∈ I d | ( λ I ) = r, m I = s } ) = { I ∈ I d | ( λ ∗ I ) = r, m I = n − s } for all r, s . Panyushev also constructed a bijection that satisfies equation (9) in [23]. That bijection is an involution,whereas ι can have large order.We list the cardinality of the sets in (10) for A in Table 2. This matrix for general n has zeros below theanti-diagonal since (8) implies ( λ ∗ I ) ≤ ( λ ( e I ) ∗ ) , hence m I ≥ ( n + 1) − ( λ ∗ I ) . With the indexing from thefirst row, the anti-diagonal entries seem to be (cid:0) n + m I n − m I (cid:1) , see A054142 from [21]. Finally, m I ≤ n + 1 − ⌈ n +1( λ I ) ⌉ TRANSITIVITY RESULT FOR AD-NILPOTENT IDEALS IN TYPE A since this is the larger possible rank of a matrix given its index of nilpotence. This explains the zeros in theupper left corner. It would be interesting to find formulas for the other cardinalities of the sets in (10), orbetter yet, the joint statistic on the two partitions λ I and λ ( e I ) attached to I .A CKNOWLEDGMENTS
We thank Vic Reiner for suggesting in a 2007 email that a statement like Proposition 6.9 should hold. Wethank Bill Casselman for directing us to Gerstenhaber’s algorithm and additional helpful comments. Wethank Tim Chow, Martha Precup, and John Shareshian for helpful conversations related to the unit intervalorder. R
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