Normality of closure of orthogonal nilpotent symmetric orbits

We study closures of conjugacy classes in the symmetric matrices of the orthogonal group and we determine which one are normal varieties. In contrast to the result for the symplectic group where all classes have normal closure, there is only a relatively small portion of classes with normal closure. We perform a combinatorial computation on top of the same methods used by Kraft-Procesi and Ohta.


Introduction
In a fundamental paper of Kostant [K], the adjoint action on a reductive Lie algebra g defined over an algebrically closed field k of characteristic 0 is studied in detail. In the course of his analysis it arose the following problem. Let A ∈ g, let C A be the conjugacy class of A and let C A be the (Zarisky) closure of C A . Is C A always a normal variety?
In his paper Kostant showed that if A is a regular nilpotent element of g, so that C A is the nilpotent cone of g, the normality is always the case.
In [H], the problem of normality of C A is reduced to the case in which A is nilpotent, possibly changing the Lie algebra g.
Later, in [B,Theorem 4.1], Broer proved the normality of a handful of nilpotent orbits in g, including the regular and the subregular orbits (if g is simple).
In the fundamental paper [KP] of Kraft and Procesi, the normality of C A is proved for every nilpotent A in the case g = gl n . Their method consists in constructing an auxiliary variety Z which is a normal complete intersection such that C A is a quotient of Z.
Kraft and Procesi extended their method to the case where g is the orthogonal or symplectic Lie algebra in [KP3]. In that case not all nilpotent classes have normal closure. They obtained necessary and sufficient conditions on the partition of A in order to have the normality of C A , under suitable hypothesis on A.
In a subsequent paper by Sommers [S3], the cases outside of this hypothesis (in [KP3]) are proved to always have normal closure. Sommers also solved the problem for the exceptional case E 6 in [S2]. Kostant and Rallis generalized the study [K] of the adjoint action on g to the action on the symmetric space in [KR]. We briefly describe their setting. Let θ be a Lie algebra automorphism of g of order 2. Then there exists a decomposition of g in θ-eigenspaces given by g = k ⊕ p where k = {A ∈ g : θ(A) = A} and p = {A ∈ g : θ(A) = −A}. Let G be the adjoint group of g and K ⊆ G be the subgroup of elements commuting with θ. We notice that k is a Lie subalgebra of g and Lie K = k. We remark that K is not necessarily connected. Moreover the adjoint action of G on g induces an action of K on g and both k, p are K-stable. The pair (g, k) is called a symmetric pair and p is the symmetric space associate to it.
In their study, Kostant and Rallis focused on the action of K on p. Among other things, they already observed that the nilpotent cone is not irreducible nor normal in general.
In [V], Vinberg further generalized the study of the adjoint action to graded Lie algebras, also called θ-groups. Here θ is an automorphism of a Lie algebra g of finite order n, thus generalizing the case n = 2 of [KR]. The graduation induced by θ, g = g 0 ⊕ · · · ⊕ g n−1 , produces an action of the θ-fixed points of the adjoint group on g 1 .
After the work of Kostant, Rallis and Vinberg, the problem of the normality of C A naturally generalizes to the case of symmetric spaces.
In [S], Sekiguchi studied deeply the geometry of the nilpotent cone in symmetric spaces. In particular he proved several results for the principal nilpotent elements (which are the analogue of the regular nilpotent elements) when the symmetric space is the orthogonal symmetric space or the symplectic symmetric space, that is, the symmetric spaces of the symmetric pairs (g, k) = (gl(n), o(n)) and (g, k) = (gl(2m), sp(m)) respectively.
The method of the auxiliary variety Z developed by Kraft and Procesi in [KP,KP2,KP3] was adapted by Ohta in [O] to the study of the singularities of the orbits in the orthogonal and symplectic symmetric spaces. In particular he proved that C A is always normal in the case of symplectic symmetric space. Moreover he found several orbits C A in the case of the orthogonal symmetric space which have a non-normal closure.
It is well known that the conjugacy classes of the nilpotent cone of the orthogonal symmetric space are parametrized by the partitions of n (see section 1 for more details). The main purpose of this paper is to give a necessary and sufficient condition on the partition corresponding to the orbit of a nilpotent symmetric element A in order to have the normality of C A . The following theorem is the main result of the paper.
Theorem 1. Let p be the symmetric space of the symmetric pair (gl(n), o(n)) and let A ∈ p be a nilpotent element. Let λ = (λ 1 , . . . , λ h ) be the partition of A. Then C A is normal if and only if with the convention that λ h+1 = 0.
The proof of theorem 1 will be carried out in section 8. The only if part was already proved in [S, O] (see section 8 for details).
We remark that, if n is even, C A is not necessarily connected, as the group K is not connected. Therefore, sometimes, C A is not normal just because it is not irreducible, being the union of two closure of SO(n)-orbits (as an example, if one takes λ = (2), one can shows that C A is the union of two lines which meet at 0). As the referee has kindly pointed out, it would be interesting to investigate the normality of the closure of the SO(n)-orbits.
We summarise the content of the rest of the paper.
In section 1 we recall some basic facts about symmetric nilpotent orbits for the orthogonal group. In section 2 we recall from [KP] the classification of nilpotent pairs via ab-diagrams and we describe the class of ab-diagrams which parametrize symmetric nilpotent pairs. In section 3 we recall the construction of the variety Z and some of its properties which are needed for the proof. In section 4 we define a condition on partitions which plays an important role in our investigation. In section 5 we introduce some combinatorial description of pairs of partitions. In section 7, resp. section 8, we prove a complete intersection, resp. normality, condition for the variety Z using a combinatorial computation carried on in section 6. I am grateful to Claudio Procesi for proposing me this problem. I would like to thank Corrado De Concini, Andrea Maffei and Paolo Bravi for many useful comments. I am really grateful to the reviewers for their precise and useful comments. A special thank goes to my PhD advisor Giovanni Cerulli Irelli for many discussions about this problem and his precious help in editing this paper.

Symmetric nilpotent orbits
In this section we introduce settings and notations for the objects studied in the paper. We follow [KP3] and [KR].
Let V be a vector space over C together with a symmetric non-degenerate bilinear form (−, −). Even if in this paper we will only be concerned with the orthogonal group O(V ), we will denote the isometry group with respect to (−, −) as G(V ) := O(V ) in order to keep the notation as close as possible with [KP3,O]. Let gl(V ) be the space of linear endomorphisms of V . Clearly G(V ) acts on gl(V ) by conjugation. We denote by D * the adjoint of an endomorphism D with respect to (−, −). The endomorphism θ : D → −D * of gl(V ) is involutive, therefore we have a decomposition into eigenspaces for θ as follows: In this paper we are concerned with the study of nilpotent orbits of G(V ) in p (V ). It is well known ([S,Sec. 3.1], [O,Sec. 0]) that the nilpotent orbits in p(V ) are completely determined by their Jordan form and every partition of n := dim V realizes a non-empty nilpotent orbit. We denote by P (n) the set of partitions of n and by C λ ⊆ p(V ) the nilpotent orbit corresponding to λ.
We recall the dimension formula for the orbit C λ from [O,Remark 8]: As the nilpotent cone of p(V ) is G(V )-stable with only finitely many orbits, we have that orbit closure C λ is G(V )-stable, and the complement C λ \ C λ is a disjoint union of finitely many orbits. The relation C µ ⊆ C λ produces a partial order on the partitions, called dominance order and denoted by µ ≤ λ, and it is given by We recall the notion of nilpotent pairs. We follow the same argument of [KP3]. Let V (resp. U ) be a finite dimensional vector space over C equipped with a non-degenerate symmetric bilinear form (−, −) V (resp. (−, −) U ). We denote L(V, U ) := Hom C (V, U ), L(V ) := L(V, V ) and we define L V,U := L(V, U ) × L(U, V ). We can interpret L V,U as the representation variety of the quiver Q = 1 2 with dimension vector n = (dim V, dim U ). This means that The group G(V ) × G(U ) acts on L V,U by change of basis. For every A ∈ L(V, U ) we define the adjoint map A * ∈ L(U, V ) as the unique map such that Lemma 2. Let A ∈ L(V, U ) be a linear map. Then Proof. By (3), we get that ker A⊥ Im A * and ker A * ⊥ Im A.
We define N V,U as the cone in L V,U of the symmetric nilpotent pairs.
As in [KP3], we have maps  [KP,Sec. 4.2] that an ab-diagram is a list of ab-strings i.e. strings with letters a and b occurring on alternate positions. As shown in [KP,Sec. 4.3], to each nilpotent pair (A, B) ∈ L V,U corresponds an ab-diagram which determines its GL(V ) × GL(U )-orbit completely; i. e. the GL(V ) × GL(U )-orbits correspond one to one to the ab-diagrams having dim V a's and dim U b's. The following picture illustrates how to associate an ab-diagram δ = δ (A,B) to a nilpotent pair (A, B): where {v 1 , · · · , v 5 } is a suitable basis of V and {u 1 , · · · , u 4 } is a suitable basis of U .
If δ is the ab-diagram of an element p ∈ N V,U , we can retrieve the Young diagram of π(p) (resp. ρ(p)) by suppressing the a's (resp. the b's) from δ. We denote by π(δ) (resp. ρ(δ)) the Young diagram obtained in this manner.
Let (A, B) ∈ L V,U be a nilpotent pair with ab-diagram δ. For every h ≥ 1, we want to infer the rank of the linear map (AB) h−1 A : V → V (resp. (BA) h−1 B : U → U ) from δ. If γ = c 1 · · · c ℓ is a string, for any 1 ≤ i ≤ j ≤ ℓ, we call c i · · · c j a substring of γ. A substring of δ is a substring of one of its ab-strings. For example, for h = 1, the rank of A equals the number of occurrences of the letter a which are not in the last position of a string; this is hence the number of occurrences of the sub-string (ab) = (ab) h in δ. With this in mind we prove the following lemma.
Notice that the map (AB) h−1 A carries v ij to u i(j+2h−1) if j + 2h − 1 ≤ ℓ i or to 0 otherwise. Therefore, rk((AB) h−1 A) equals the number of pairs (i, j) such that c ij = a and j + 2h − 1 ≤ ℓ i . Moreover, every such pair (i, j) corresponds one to one to the substring c ij . . . c i(j+2h−1) of δ, which is the substring (ab) h . A similar argument computes rk((BA) h−1 B).

Ortho-symmetric ab-diagrams
In this section we give a combinatorial description of the ab-diagrams of the elements of The automorphism σ of G ′ defined by has the symmetric pairs as the set of fixed points. It is straightforward to check that all hypothesis of [MWZ,Prop. 2.1] hold. Thus we get that: Let δ be an ab-diagram associated to an orbit G ′ .Y . We say that δ is ortho-symmetric if We are left with the following problem: which ab-diagrams are ortho-symmetric?
We call a symmetric pair (A, A * ) indecomposable if it can not be written as a direct sum of two nontrivial symmetric pairs. We claim that the ab-diagrams of indecomposable nilpotent symmetric pairs are given in table 1. This table closely follows table II  Proposition 4. An ab-diagram δ is ortho-symmetric if and only if it is a disjoint union of finitely many ab-diagrams from table 1.
Proof. We say that an ab-diagram η has property P if, for every h ≥ 1, the number of occurrences of (ab) h in η equals the number of occurrences of (ba) h in η.
Every ab-diagrams α k , β k , ε k in table 1 has property P. Indeed in the diagrams α k and β k , for every h ≥ 1, there are exactly k − h + 1 occurrences of (ab) h and of (ba) h . The diagram ε k has property P by symmetry of its two ab-strings.
Let (A, A * ) ∈ N V,U be a nilpotent symmetric pair with ab-diagram δ, an ortho-symmetric ab-diagram. For every positive integer h, we notice that the pair ((AA * ) h−1 A, (A * A) h−1 A * ) is nilpotent symmetric. By lemma 2, we have: which, by lemma 3, implies that δ has property P.
We claim that an ab-diagram δ with property P is a disjoint union of finitely many abdiagrams from table 1. We proceed by contradiction: suppose that there exist ab-diagrams with property P which are not disjoint union of finitely many ab-diagrams from table 1 and take δ one with the least number of ab-strings. Let s be one of the ab-strings in δ with maximal length and let l be its length.
If l = 2k + 1 is odd, s = α k or s = β k . Let δ ′ be the ab-diagram such that δ = s ⊔ δ ′ (we cannot have δ = s, otherwise δ is already in table 1).
If l = 2k is even, without loss of generality, we assume that s starts with a. Then, in s there exists exactly one occurrence of (ab) k and no occurrence of (ba) k . As δ satisfies property P, by maximality of l, there must be an ab-string s ′ in δ of length l = 2k starting with b. Thus s ⊔ s ′ = ε k , so let δ ′ be the ab-diagram such that δ = s ⊔ s ′ ⊔ δ ′ (again, it cannot be empty).
In both cases, δ and δ \ δ ′ have property P, therefore also δ ′ has property P. By minimality of δ, δ ′ is disjoint union of finitely many ab-diagrams from table 1, thus so is δ. This contradicts the existence of such a δ and concludes the only if part of the proposition.
On the other hand, we show that every ab-diagram in table 1 is ortho-symmetric. In order to show this, we just need to construct a symmetric pair (A, A * ) associated to each diagram α k , β k , ε k . α k : Let D : C k+1 → C k+1 be a symmetric nilpotent endomorphism of nilpotent order exactly k + 1. Let D = I • X be the canonical decomposition of the map D through its image D(C k+1 ), so that X : C k+1 → D(C k+1 ) and I : D(C k+1 ) ֒→ C k+1 . Then D induces a non-degenerate bilinear symmetric form in D(C k+1 ), as shown in [KP3,4], and X = I * .
As dim D(C k+1 ) = k and both X, I have maximal rank, we immediately get that the ab-diagram of the symmetric nilpotent pair (X, I) is α k . B), so that it is a nilpotent pair between the dual spaces (V * , U * ) with ab-diagram ba · · · a. Then we can equip V ⊕ V * and U ⊕ U * with the non-degenerate symmetric bilinear form given by the duality pairing. Therefore (A, B) ⊕ (B * , A * ) is a symmetric nilpotent pair and its ab-diagram has type ε k .

The variety Z
For each partition λ of n we are going to construct a variety Z (λ) (or just Z, if the partition λ is clear from the context) with the following property: there exists a quotient map Z → C λ . In this way, we will be able to assert some properties of C λ by proving them for the variety Z. The construction of Z is analogous to the one carried out in [KP3] and it has already been described in [O]; here we just recall its definition. Let λ ∈ P (n) be a partition and let t = λ 1 be the number of columns of λ. As in [KP3], we define a variety Z using the representations of the quiver: x y x y x y Let n i = t j>i λ j so that n 0 = n and n t = 0. We fix the dimension vector n = (n 0 , . . . , n t ) and vector spaces V i such that dim V i = n i . We also fix a symmetric bilinear non-degenerate form on each V i . The variety Z consists of the representations of dimension vector n of the quiver Q t with relations: • xy = yx; • y = x * (the adjoint map as in (3)).
Therefore, each point z ∈ Z is a sequence of maps As in [KP3], we can think of Z as a schematic fiber in the following way. Let then we put Z := Φ −1 (0). As M is an affine space, this interpretation will be useful in section 7 to show that Z is a complete intersection variety (at least for some λ).
As in [KP3], let G = G(V 0 ) × · · · × G(V t ) be the group which acts on Z by change of basis and By the same argument as in [KP3], we have the following: Proposition 5. The map Θ splits through the quotient by the group H as (In particular, Θ −1 (C µ ) is stable under the action of G(V 0 ) for each partition µ ≤ λ.) By restricting the quiver Q t to two consecutive vertices i − 1, i, we get a map Z → N Vi−1,Vi . As all the orbits in N Vi−1,Vi are defined by their ortho-symmetric ab-diagram, we can map each point z ∈ Z and i = 1, . . . , t to its i-th ab-diagram τ i . Therefore, for each string of ab-diagrams The following are necessary conditions on τ in order to have Z τ = ∅: 1. τ i must have exactly n i−1 a's and n i b's, 2. ρ(τ i ) = π(τ i+1 ) for i = 1, . . . , t − 1, 3. each τ i must be ortho-symmetric.
We will denote by Λ (λ) (or, shortly, by Λ) the set of all strings τ which verify the three conditions above.
As in [KP, Lemma 5.4], [KP3,8.2], Θ −1 (C λ ) is given by exactly one stratum Z τ 0 . The string of ab-diagrams τ 0 can be constructed in the following way: for all i = 0, . . . , t − 1, let λ i be the partition of n i given by the columns of λ with indices greater than i; then let τ i+1 be the ab-diagram built by setting one a on each box in the Young diagram of λ i and by setting one b between each pair of consecutive a's.
Example. Let λ = (3, 1) be a partition. Then the string of ab-diagrams τ 0 is the following: The stratum Z τ 0 is open in Z as it is the subvariety of the points We have the following key result after the manner of Kraft, Procesi and Ohta.
We recall a dimension formula for the strata Z τ due to Otha [O,Prop. 6]. First, for each ab-diagram δ, we denote by a i (resp. b i ) the number of rows of δ starting by a (resp. b) of length i. We define ∆(δ) := Proposition 7. [O,Prop. 6] Let τ = (τ 1 , . . . , τ t ). We have In the following sections we focus on the combinatorics of the τ ∈ Λ. We need tools to effectively manipulate formula (4), in particular we aim at proposition 13 and proposition 17 (see section 6).

s-step condition
In this section we define a condition which plays a pivotal role in the subsequent study of the nilpotent symmetric orbits.
We want to highlight some simple properties of this definition. First of all, s 1 -step condition implies s 2 -step condition for each s 2 ≥ s 1 . Secondly, if λ satisfies the s-step condition, then if we remove the first row or the first column from λ, the resulting partition still satisfies the s-step condition.

Differences between partitions
In this section we study some quantitative ways to address the difference between two partitions. Let λ, µ ∈ P (n) be two partitions of n such that λ > µ.
We can check that in this case q(λ, µ) is the number of gray boxes or the number of black boxes. This is an easy fact (it is not even needed that λ > µ) that can be derived from the following lemma.
Moreover, both the following hold: q(λ, λ 1 ) = 1 and q(λ 1 , µ) = q − 1. We apply induction on the pair (λ 1 , µ) and we conclude the proof of the first part of the lemma.
Remark. In view of lemma 10 we can interpret q(λ, µ) as the minimum number of boxes needed to lower in order to obtain µ from λ.
Moreover, (11) holds strictly if there exists a column index c such that µ c > λ c + 1 or a row index r such that µ r > λ r + 1.
We start by noticing that the s-step condition implies that, for every pair of row indices r 1 < r 2 , λ r1 − λ r2 ≤ s(r 2 − r 1 ).
Fix r 1 to be an index row such that λ r1 ≥ µ r1 and fix r 2 to be an index row such that λ r2 ≤ µ r2 . Lemma 10 asserts that λ i rj is a monotone sequence for both j = 1, 2; therefore, for such r 1 , r 2 , we get λ i r1 − λ i r2 ≤ λ r1 − λ r2 , so we get As q(λ i , λ i+1 ) = 1, we already observed that the Young diagrams of λ i , λ i+1 have only one box in different positions. This means that there exist exactly two columns c i,1 < c i,2 which are not equal between λ i , λ i+1 . Similarly, there exist exactly two rows indices r i,1 < r i,2 .

In a similar fashion, let
If there exists r such that µ r > λ r + 1, then |I rows (r)| ≥ 2 and we take j ∈ I rows (r) to be the minimum and i ∈ I rows (r) to be another index. We have λ i ri,2 > λ j ri,2 = λ ri,2 , so we get λ i ri,1 − λ i ri,2 < λ ri,1 − λ ri,2 ≤ s(r i,2 − r i,1 ).
In both cases (13) holds strictly for at least an i. Therefore (12) holds strictly for this i, and (11) must hold strictly as a consequence. Thus, the lemma is proven.

Inequalities on the dimensions
In this section we will use the tools introduced in section 5 to effectively compute the dimensions of the strata in Z.
Let λ = (λ 1 , . . . , λ k ) be a partition of n := |λ| with t := λ 1 columns. We want to compare the dimension of Z τ 0 with the dimension of each other stratum Z τ with τ ∈ Λ (λ) . Let µ = π(τ ), so that µ ≤ λ. The aim of this section is to give some sufficient combinatorial conditions on the pair of partitions (λ, µ) in order to secure a bound on the difference dim Z τ 0 − dim Z τ .
The aim of the current section is to prove proposition 13 and proposition 17. In order to introduce the stronger inequality in proposition 17, we need an in-depth study of the combinatorics of Λ, so we prefer to introduce it later.
For each real number x, we have that: The proof will proceed as follows: we take the dimension formula of the strata of Z given by (4) then we will estimate each term on the difference. Each of the following lemmas deals with one of the terms.
The conditions τ ∈ Λ and π(τ 1 ) = µ place important combinatorial restrictions in the choice of the ab-diagrams τ 1 , . . . , τ t . We are going to describe them in the following paragraphs.
Let Z (µ) be the variety Z built from the partition µ (rather than the partition λ). We will denote by σ 0 the unique stratum in Λ (µ) such that Θ(Z (µ) σ 0 ) = C µ . As in the case of τ 0 , every ab-diagram σ 0 i has only rows starting and ending by a, in particular they all have odd length, so: The sequence of ab-diagrams σ 0 does not belong to Λ (λ) because the integers n i computed from µ are different from those computed from λ.
For each i = 1, . . . , t, let d i be the list of a's and b's we need to add to σ 0 i in order to obtain τ i . Let d a i (resp. d b i ) be the number of a's (resp. b's) in d i . We remark that d a i and d b i depend only on (λ, µ) and not on the particular τ . In fact, we can compute d a i by taking the differences and therefore we have Example. Let τ ∈ Λ (λ) and let σ 0 ∈ Λ (µ) . Suppose that the i-th diagrams are the following: Then aug δ (d) has three elements: Lemma 15. Let δ 0 be an ab-diagram with rows starting and ending only with a. Let d be a list of d a a's and d b b's. Let δ ∈ aug δ 0 (d). Then the following holds: Proof. The ab-diagram δ is obtained from δ 0 by adding d a a's and d b b's. Let L (resp. S) be the set of rows of δ longer than 1 (resp. of length 1) built using only letters in d. As o(δ) counts some of the rows in δ, we compare the difference o(δ) − o(δ 0 ) with the number of rows built only with letters in d and we get o(δ) − o(δ 0 ) ≤ |L| + |S|.
Let S a (resp. S b ) the rows of S starting with a (resp. b), so that S = S a ⊔ S b . Recall that a i (resp. b i ) is the number of rows of length i starting with a (resp. b); in this particular ab-diagram δ we have |S b | = b 1 and |S a | ≤ a 1 .
If |S a | ≥ |S b |, then |L| + |S a | ≤ d a i because each row in L or in S a contains at least one a. If the opposite holds true, then |L| + |S b | ≤ d b i for the same reason. Thus the lemma is proven. We can introduce a small, but very important, improvement on lemma 15.
Lemma 16. In the same setting of lemma 15, if we furthermore assume that d = (b), that is d a = 0 and d b = 1, and that δ 0 has l rows of length 1, we have the stronger equality: Proof. As all the rows in δ 0 have odd length, we cannot extend exactly one row with letter b, otherwise the resulting ab-diagram would not be ortho-symmetric. So we must place b on a new row.
Using this lemma we will be able to introduce the previously announced stronger version of proposition 13, namely: Proposition 17. Let λ ≥ µ be two partitions and let Z = Z (λ) be the variety built from λ. Let Z τ 0 be the only stratum of Z with π(τ 0 1 ) = λ and let Z τ be a stratum with τ = (τ 1 , . . . , τ t ) such that π(τ 1 ) = µ. As in proposition 13, we suppose that there exists a real number x such that: If there exists an index i such that d i = (b) and σ 0 i has l rows of length one, then: The proof of proposition 17 will be given together with the proof of proposition 13 at the end of this section.
We want an estimate of the sum of the terms max{d a i , d b i } depending only on λ, µ.
Lemma 18. Let λ ≥ µ be partitions, so that we can define d a i , d b i for all i = 1, . . . , t. Then Proof. First we prove the claim for q(λ, µ) = 1. In this case, µ is obtained from λ by moving down a single box B. Let us call c 1 (resp. c 2 ) the column where B lies in µ (resp. λ); so c 1 < c 2 . By (15), one gets d a i = 1 for each c 1 < i ≤ c 2 , and 0 otherwise. Moreover one also gets that d b i = 1 for each c 1 ≤ i < c 2 and 0 otherwise. Therefore max{d a i , d b i } = 1 for all c 1 ≤ i ≤ c 2 and 0 otherwise. Therefore and the conclusion holds in this case.
We are finally able to prove both proposition 13 and proposition 17.
Proof of proposition 13 and proposition 17. We start from the dimension formula of the strata (4). We recall that ∆(τ 0 ) = 0 as no ab-diagram of τ 0 has rows starting with b. Therefore In order to obtain the sharper result of proposition 17, it is enough to use lemma 16 in place of lemma 15 in the second to last step.

Complete intersection conditions
In this section we want to give condition under which we can make sure that Z is a complete intersection variety.
Proposition 19. Let λ be a partition and recall that Z = Φ −1 (0) ⊆ M . We have Proof. In order to prove the equality of the required dimensions, we will work with the dimension formula given by (4). We are going to prove that We immediately have: On the other hand, we have: We have that π(τ 0 1 ) = λ; we then use formula (2). We also have that o(τ 0 i ) is precisely the number of rows of τ 0 i (as they all have odd length), the number of rows of τ 0 i is equal to the number of rows of λ i := π(τ 0 i ), the partition λ i has clearly λ i 1 rows and λ i 1 = λ i , that is the i-th column of λ. Finally, we have that ∆(τ 0 i ) = 0, because there are no rows in τ 0 i starting with b. Therefore: we recall that λ i = n i−1 − n i and that n 0 = n = |λ|, n t = 0; so We can proceed to the main result of this section.
Proposition 20. If λ satisfies the 2-step condition, then Z is a complete intersection variety.
The argument is basically the same used in [KP, Theorem in 3.3] and [KP3,Theorem in 5.3]. Here we merely put the pieces together. One of these pieces is the following technical lemma.
Lemma 21. If the partition λ satisfies the 2-step condition and τ ∈ Λ is a string of ab-diagrams different from τ 0 , then dim Z τ 0 > dim Z τ .
Proof of proposition 20. We start recalling proposition 6 and proposition 19. As Z \ Z τ 0 consists of finitely many strata Z τ and, by lemma 21, codim Z (Z τ ) ≥ 1, we deduce that Z = Z τ 0 . That implies that Z is a complete intersection variety smooth in codimension 0 and Z is reduced as a scheme.
We can finally use lemma 12 and propositions 13 and 17 to prove lemma 21.
Proof of lemma 21. Let µ = π(τ ), so in particular µ < λ, as τ = τ 0 . Therefore lemma 12 implies 2r(λ, µ) ≥ c(λ, µ) + q(λ, µ) and, by combining it with proposition 13 we immediately get So our concern is reduced to obtain a strict inequality. We will gain the strict inequality either by proving that (19) holds strictly for the pair (λ, µ) or by using proposition 17 with l > 0. Let c be the first column such that µ c > λ c and let r = µ c . This means that the first c − 1 columns of λ and µ are the same and µ r > λ r . On the basis of c and r, we distinguish three cases.

Normality conditions
In this section we prove theorem 1. We start by proving the necessary condition as a consequence of the works by Ohta and Sekiguchi.
Theorem 22 (the only if part of theorem 1). Let λ be a partition of n and suppose that λ does not satisfy the 1-step condition. Then C λ ⊆ p is not normal.
The degeneration µ ′ < λ ′ is obtained from µ < λ by removing the common rows and columns (of µ and λ). By theorem [O,Theorem 2], the singularity C µ of C λ is normal if and only if the singularity C µ ′ of C λ ′ is normal.
Moreover, by construction, C λ ′ is the variety of the principal (or regular) nilpotent elements of p (in dimension m); therefore we also have that C λ ′ = N (p), that is the cone of the nilpotent elements in p. As µ ′ is the minimal degeneration of λ ′ , C µ ′ is the variety of subregular nilpotent elements of p. The theorem [S, §3.1, Theorem 7.] proves that the singularity C µ ′ is not normal in N (p). So this concludes the only if part.
We turn to the sufficient condition. We will make use of the construction of the variety Z.
Theorem 23 (the if part of theorem 1). If the partition λ satisfies the 1-step condition, then Z is a normal variety. In particular C λ is a normal variety.
As λ satisfies the 1-step condition, also λ satisfies the 2-step condition, so, by proposition 20, Z is a complete intersection variety. At this point the proof of theorem 23 is a direct consequence of the following lemma, as in [KP, Section 3.7].