An algebra of commuting nilpotent matrices

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1 An algebra of commuting nilpotent matrices Anthony Iarrobino Department of Mathematics, Northeastern University, Boston, MA 02115, USA.. Talk at IMAR Bucharest, July 3, 2008 Work joint with Roberta Basili. Abstract: Let Mat n (K) denote the ring of n n matrices over a field K. Fix a nilpotent n n matrix B of Jordan partition P, and consider the centralizer C B of B, and its subvariety N B of nilpotent matrices. Denote by N 2 (n, K) the variety of commuting pairs of nilpotent matrices. We describe recent work on both these varieties, and the connections with previous work by J. Briançon et al on the fibre H [n] of the punctual Hilbert scheme Hilb n (P 2 ) of the plane over a point p P 2. R. Basili defined a maximal nilpotent subalgebra U = U B of N B. We describe an involution on C B, and give bases for the quotients U i /U i+1. 1

2 References: V. Baranovsky: Transform. Groups 6 (2001), no. 1, 3 8. R. Basili: J. Alg 268 (2003), R. Basili-I. ArXiv.AC: , J. Algebra(2008). Bas-I2: Involution (2007) T. Košir-P. Oblak ArXiv.AC: , to appear, Transf. Groups P. Oblak, Arxiv math.ac/ , LAMA (2007) D. I. Panyushev: JPAA 212 no. 4 (2008), A. Premet: Invent. Math. 154 (2003), no. 3, Note. The natural connection between commuting n n nilpotent matrices and the fibre of the punctual Hilbert scheme Hilb n (A 2 ) over a point p of A 2 was noted by H. Nakajima; and used by V. Baranovsky, R. Basili, and A. Premet, to study the irreducibility of the variety of pairs of commuting nilpotent matrices [Bar2001], using J. Briançon s work, or vice versa. Since a pair of commuting matrices may not have a cyclic vector, the theory of pairs and triples of commuting nilpotent matrices is related to that of Hilbert schemes, but is not isomorphic. 2

3 Acknowledgment. Our work on the algebra U B was greatly assisted by conversations at the Halifax conference: Combinatorial Algebra meets Algebraic Combinatorics in January 2008, among R. Basili, T. Košir, J. Weyman, and I. We are grateful to discussions with J. Emsalem, B. A. Sethuraman, M. Boij while at Mittak Leffler Institute 2007, to J. Bernik. P. Oblak and T. Košir at LAW 08, and to work of T. Harima and J. Watanabe. Preface: Consider a subalgebra A = A S,T Mat n (K), K a field, comprised of nilpotent matrices A, and defined by a set S (zeroes of entries), and by equalities T among entries. i. S [1, n] [1, n] A ij = 0, for (i, j) S. ii. T = (T 1,..., T t ), T i S c = [1, n] [1, n] S such that for each i, 1 i t, A uv = A u v when (u, v), (u, v ) T i. Lem 0.1. A and each A k, k N are irreducible sets. Question 1. What is the rank of the general element of A k? 3

4 Question 2. Let A be a generic in A. What is rank A k? Question 2. What is the partition P A determined by the Jordan blocks of A for a generic A A? When T =, these were answered differently by R. Gansner, and S. Poljak in terms of the digraph D(A) = def D(A) for a generic A A (i.e. D(S c )). Definition 0.2. Digraph D(A) of a matrix A M n (K): Directed graph: Vertices = {1, 2,... n}; An arrow from i to j iff A ij 0. Def. Two k-walks W = (w 1,..., w k ), W = (w 1...., w k ) on D are vertex independent if for each i, 1 i k, w i w i. Thm 0.3. [Ga, Pol] Assume T =. i. (Gansner) Consider the sequences C = (c 1, c 2,...), c i = max # distinct vertices covered by i chains of D; 4

5 D = (d 1,..., ), d i = max # vertices covered by i antichains. Then for A generic in A, P (A) = C, P (A) = D. ii. (Poljak) The maximum rank of A k, A A, and also of A A k is the maximum number of (vertex) independent k-walks in the digraph D(A). For (i.) see also T. Britz and S. Fomin [BrFo]. For (ii.) see also H. Knight and A. Zelevinsky [KnZe] Open: Find rank A k concisely for specific algebras A. Problem: Generalize Poljak and Gansner s Thm. to T. Ex 0.4. Consider the algebra A S Mat 5 (K): here the zero entries are S, starred { } entries form S c, and T =. 5

6 A S : A = [The next page is handwritten digraph for this A, after P. Oblak [Ob2]] 6

7 1 What is Q(P ), maximum nilpotent orbit in N B? Let K = algebraically closed field, Mat n (K) = n n matrices. N n (K) = {nilpotent A Mat n (K)}. Fix B N n (K) Jordan, of partition P B = (λ 1 λ t ). C B = A Mat n (K) [A, B] = 0. N B = C B N n (K). Problem 1.1. Find Q(P ) = {Jordan partitions of A N B }. Ex 1.2. P = (4), so B is regular (single Jordan block) a b c a b B =, A = a When a 0, A 3 0 and P A = (4). When a = 0, b 0, A 2 = 0, rka = 2, P A = (2, 2) When a = b = 0, c 0, then P A = (2, 1, 1). When a = b = c = 0 then P A = (1, 1, 1, 1). (3, 1) / Q(P ) for P = (4). 7

8 1.1 The morphism π : C B C B. (C B semisimple) R. Basili [Bas1] using [TurAi] parametrized N B, and U B Ex 1.3. Let P = (3, 3, 2), B = J P. Then A C B satisfies: a 1 11 a 2 11 a 3 11 a 1 12 a 2 12 a 3 12 a 1 13 a a 1 11 a a 1 12 a a a a a 1 21 a 2 21 a 3 21 a 1 22 a 2 22 a 3 22 a 1 23 a 2 23 A = 0 a 1 21 a a 1 22 a a a a a 2 31 a a 2 32 a 3 32 α33 1 a a a α33 1 with entries in the ring Z[a 1 11,..., a 2 33] in 22 variables. Let J = Jacobson rad. of C B, C B = C B/J semisimple quotient. Set A(3) = a1 11 a 1 12 a 1 21 a 1 22 Morphism: π : C B C B, A(2) = (α 1 33), : A (A(3), A(2)). Note: N B = π 1 (N 2 (K), 0). J = π 1 (0, 0). 8

9 Let U B = π 1 (( 0 a ), 0 ) Let P = (p r 1 1,..., pr s s ), p 1 > > p s. Mat r = Mat r1 (K) Mat rs (K),, nilp. subalgebra of N B. N r = N r1 (K) N rs (K). π : C B C B. Lem 1.4. [Bas1]: N B = π 1 (N r ) and is irreducible. Def. Denote by Q(P ) the partition of a generic A N B. 1.2 Maximal nilpotent subalgebra U B of C B. Let U r = U r1 (K) U rs (K) (s.u.t.). Let U B = π 1 (U r ). Lem 1.5. U B is a maximal nilpotent subalgebra of C B. Each element of N B is similar to an element of U B under the conjugation action of C B. Cor 1.6. Q(P ) is the partition of a generic element of U B. Warning. There is no simple analogue of Lemma 1.5 for pairs A, A N B. We denote by D(P ) the digraph of a generic A in U B. 9

10 Lem 1.7. D(P ) has no loops. If A U B is generic then k N, i, j 1 i, j n, (A k ) ij = 0 (A k+1 ) ij = 0. Question 3. Is the rank of A k, k = 1, 2,... an invariant of D(P )? Is this rank the same as that for a generic matrix of zeros and variables with the same digraph [Pol, KnZe]? (Can we ignore the equalities among entries in finding Q(P )?). 1.3 What we know about Q(P ) brief look. Def. A string or almost rectangular subpartition of P is one s.t. largest - smallest part 1. Let r P = minimum # strings needed to write P. Ex. P = (6, 6, 5, 4) = (6, 6, 5) (5, 4), so r P = 2. Thm 1.8. (Basili [Bas2]): Q(P ) has r P parts. Def. The index of a partition Q is its largest part: So index Q(P ) = 1 + max{k A k 0}. Let S P = set of parts of P and n i the multiplicity of i in P. 10

11 Let s i = k>i n k and let j i = max{n i 1 + n i, n i + n i+1 }. Thm 1.9 (Index of Q(P )). (P. Oblak ([Ob2], later [BaI2]) Let K be an infinite field. The index of Q(P ) satisfies, index(q(p )) = max 1 i p1 {2s i + n i + (i 1)j i }. (1.1) Thm R. Basili-I [BaI1], D. Panyushev[Pan]. Q(P ) = P the parts of P differ pairwise by 2. (BI: P stable if Q(P ) = P. Panyushev: P self large ). The Hilbert function H of the commutative algebra A A,B = K[A, B] gives the dimension in each degree of the associated graded algebra. For a codimension two algebra, the partition P (H) is dual to the graph of H (to {h i, i = 0, 1,... }) Ex. For H = (1, 2, 3, 3, 2, 2, 0), P (H) = (6, 5, 2). Thm 1.11 (Pencils). [BaI1] Suppose A U B, let H = H(K[A, B]) and let K be alg. closed, char K = 0. Then for generic λ P 1 the Jordan block sizes of the action of A + λb on K[A, B] are given by the parts of P (H). 11

12 Thm (T. Kosir and P. Oblak)[KO] Q(Q(P )) = Q(P ). (Q(P ) is stable ). Proof. Show A = K[A, B] is Gorenstein if A N B is generic, extending a result of V. Baranovsky that A has a cyclic vector [Bar2001]. F.H.S. Macaulay characterized the Hilbert function H(A), for A Gorenstein: H drops by at most 1 ( i, h i h i+1 1). Then the dual partition P (H) is stable that is, the parts of P (H) differ pairwise by at least 2. Ex. H = (1, 2, 3, 4, 3, 3, 2, 2, 2, 1) has P (H) = (10, 8, 4, 1). Ex P = (3, 1). Take A generic in N = U B = π 1 (0, 0) a b f a 0 B =, A = d 0 A 2 = α e 13, α = a 2 + df. If α 0, P A = (3, 1) (P is stable). When α = 0, P A = (2, 2) or (2, 1, 1) or (1, 1, 1, 1). 12

13 Ex P = (3, 1, 1). U B = π 1 (0, ( 0 c 0 0 )) a b e 1 e a 0 0 B =, A = 0 0 f 2 0 c 0 0 f U B. Here A 3 = (ce 1 f 1 ) e 13, A 4 = 0, so (A generic) Q(P ) = (4, 1). Also, A 3 = 0 iff P A (3, 1, 1), Note: the C B orbit of (3,1,1) in U B is reducible, though its C B orbit in N B is irreducible. We have 0 0 a 2 + β 0 ce 1 A 2 = 0 0 cf 1 0 0, A 3 = 0 0 cdf 0 0, A 4 = 0. where β = e 1 f 2 + e 2 f 1. 13

14 When ce 1 f 1 0 we have P A = (4, 1) = Q(P ) When ce 1 f 1 = 0 but ce 1 or cf 1 or α 0 we have rank A 2 = 1, and P A = (3, 2) if ranka = 3 or (3, 1, 1) if rank A = 2. When A 2 = 0, P A = (2, 2, 1), (2, 1, 1, 1) or (1, 1, 1, 1, 1). Q(P ) = (4, 1) = {(4, 1), (3, 2), (3, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1)}. Thm. P. Oblak [Ob1] characterizes Q(P ) for commuting pairs (A, B) satisfying AB = 0. Remark. The Hilbert scheme Hilb n (K[x, y]/(xy)) is a connected set of lines, and is reducible (folklore 1 ). This is an example of the following result, which merits more notice: Thm. (A. Shoshistvili [Sh], 1976). Let B be any local ring. Then Hilb n (B) is P 1 connected. Cor. The family of commuting pairs of nilpotent matrices satisfying a finite set of polynomial conditions, and having a cyclic vector, is connected by lines P 1. 1 This is easily checked, and was remarked to me by S. Kleiman and D. Eisenbud). 14

15 2 The algebra U B. Let B be Jordan of partition P. Then U B, a maximal subalgebra of N B, has a filtration U B U B 2... U B i(q(p )) = 0. Definition 2.1. Pow(P ) ij = k > 0 if both (A k ) ij 0 and (A k+1 ) ij = 0 for A generic in U B. Pow(P ) ij = 0 if A ij = 0. M X1 = the matrix whose nonzero entries are those of A for which Pow(P ) ij = 1, and whose other entries are zero. Powxe(P ) = M X1 + (M X1 ) Results i. Bases for (U B ) i /(U B ) i+1. ii. An involution on C B and the POS D P, whose restriction to U B gives symmetries among the bases in (i). iii. Algorithm to construct matrices Pow(P ), Powxe(P ) closely related to the digraph D(P ). 15

16 0 a b e 1 e a 0 0 Ex 2.2. Let P = (3, 1, 1), A = generic in U B. 0 0 f 2 0 c 0 0 f a 2 + β 0 e 1 c A 2 =, β = e 1 f 2 + e 2 f f 1 c 0 0 A 3 = e 1 f 1 c 2 E a 0 e a 0 0 Pow(P ) =. M X1 (P ) = c f Powxe(P ) = M X1 (P )+(M X1 (P )) 2 + = 0 a ce 1 f 1 + a 2 f 1 cf a ce 1 0 c 0 0 e

17 Interpretation of Powxe(P ), Pow(P ). Ex 2.3. Let P = (3, 1, 1). Recall M X1 (P ) = Powxe(P ) = M X1 (P )+(M X1 (P )) 2 + = 0 a 0 f a c 0 0 e a ce 1 f 1 + a 2 f 1 cf a ce 1 0 c 0 0 e Remark 2.4. The monomials in the (i, j) entry of Powxe(P ) correspond to maximal paths from i to j, in the sense that there is no longer path from i to j properly containing the given one. Pow(P ) gives the (top) degree of each entry of Powxe(P ). There is an identification of edges corresponding to Toeplitz equalities in small blocks (so a set T 0). Thm 2.5 (Basis Theorem). A set of basis vectors for (U B ) k mod U B ) k+1 are given by the vectors that correspond to the entries k of Pow(P ) mod Hankel relations: the basis has one vector v h for each Hankel small diagonal h whose entries equal k: v h = (i,j) h e ij 17

18 A basis for U B ) k is given by those vectors as above corresponding to the small Hankel diagonals of Pow(P ) whose entries are at least k. 18

19 Ex 2.6. P = (3, 1, 1). 0 x 12 x 13 x 14 x x A = 0 0 x 43 0 x x , Pow(P ) = Here v 11 = e 12 + e 23, v 12 = e 14, v 13 = e 45, v 14 = e 53 ; v 21 = e 15, v 22 = e 43, v 3 = e 13 U B /U 2 B = V 1 = v 11, v 12, v 13, v 14. U 2 B /U 3 B = V 2 = v 21, v 22 and U 3 B = V 3 = v 3. The action of ι extends to V, and each V i is ι-invariant. Remark. There is symmetry here and for some other (not all) P in the U B -Hilbert functions, when stratified by large matrix blocks, corresponding to 3, (3, 1), 1. Here H UB (V 1 ) = (1, 2, 1), H UB (V 2 ) = (0, 2, 0). Problem. Let A i = generic element of U B i. We have, evidently, rank A i rank A i. Compare these ranks. 19

20 The algebra U B for B = J P, P = (3, 1, 1). B = , A = 0 a b e 1 e a f 2 0 c 0 0 f 1 0 0, Pow(P ) = We give the following basis for U B : v 11 = v 21 = v 12 = v 22 = v 13 = v 3 = v 14 =

21 Remark. We have (U B ) 3 = v 3, (U B ) 2 = v 21, v 22 ; v 3, U B = v 11, v 12, v 13, v 14 ; v 21, v 22 ; v 3. The involution σ, a generalized transpose (we ll define it later) satisfies σ(v 12 ) = v 14, σ(v 21 ) = v 22 and leaves the other basis vectors fixed. The nonzero multiplications among the basis vectors are v 13 v 14 = v 22. σ(v 13 v 14 ) = σ(v 14 ) σ(v 13 ) = σ(v 22 ), so v 12 v 13 = v 21. Also, v 12 v 22 = v 3, and, applying σ, v 21 v 14 = v 3. Also v 11 v 11 = v 3. 21

22 Pairs of commuting matrices in U B - an example. Question 3. Is {(A, C) A, B U B, [A, B] = 0} irreducible? (Similar question for A, B N B ). We conjecture the answer NO in general for U B and N B. D.I. Panyushev 2 NO: for C B, P = (3 4, 1 2 ). Answer for B = J P, P = (3, 1, 1) and U B : YES. Use that the eigenspaces V + (symmetric) and V (antisymmetric) for σ satisfy, [A, B] V when A, B are both in V + or both in V. [A, B] V + when one of A, B in V + and the other in V. ( ) ( ) Since B = v 11 and B 2 = v 3 commute with everything in U B we may reduce to v, v each having zero component on v 11, v 3. Among c = v 13, w 1 = v 12 + v 14, w 2 = v 21 + v 22 V + ; and u 1 = v 12 v 14, u 2 = v 21 v 22 V only the bracket [w 1, c] = u 2 is nonzero. Thus, for v = a w 1 + b c + c w 2 + d v 1 + e v 2, and v = a w 1 + b c + c w 2 + d v 1 + e v 2 we have [v, v ] = (ab a b) v 2, so [v, v ] = 0 ab a b = 0. This is an irreducible condition on the coefficients of v, v on the chosen basis! 2 Bus-ride lemma that D.I. Panyushev showed at LAW 08. His proof uses Gl 2 action to show there is a high-dimensional component of nilpotent pairs 22

23 3 An involution on partitioned matrices Ex 3.1. The involution σ s (2, 3) takes M 5 (R) M 5 (R): a b α 4 α 5 α 6 c d α 1 α 2 α 3 α 3 α 6 e f g α 2 α 5 h i j α 1 α 4 k l m to d b α 4 α 5 α 6 c a α 1 α 2 α 3 α 3 α 6 m j g α 2 α 5 l i f α 1 α 4 k h e. Definition 3.2. The action of σ s (a, b) on M a+b (R) : i. reflects the entries in the a a block at the upper left, and in the b b block in the lower right, about their non-main diagonals. ii. Sends the b a block in the lower left into the a b block at upper right by transpose followed by reversing the order of rows, then reversing the order of columns. 3.1 An involution on the POS D P and D P, P n. Let P = (..., i n i,...) be a partition of n = i i n i; S P = {i n i 1}. Label the vertices V = (1,..., n) of the digraph: (i, j, k), i S P, 1 j i, 1 k n i. We define σ : V V σ(i, j, k) = (i, i + 1 j, n i + 1 k). (3.1) 23

24 On edges we define σ ((i, j, k), (i, j, k )) = (σ(i, j, k ), σ(i, j, k)). (3.2) Ex 3.3. [σ for P = (3 2, 1 3 )] Here n = 9. On vertices v, we have v 3, 1, 1 3, 2, 1 3, 3, 1; 3, 1, 2 3, 2, 2 3, 3, 2 1, 1, 1 1, 1, 2 1, 1, 3 σ(v) 3, 3, 2 3, 2, 2 3, 1, 2; 3, 3, 1 3, 2, 2 3, 1, 1 1, 1, 3 1, 1, 2 1, 1, 1 For P = (3 2, 1 3 ). A A = Mat(D P ) (take T = ) 0 a 1 a 2 d d 2 d 3 f 4 f 5 f 6 0 a 3 a 4 d d 2 d 3 e 4 e 5 e a 1 0 d d a 3 0 d d d d c c 2 0 a 3 a 4 f f 2 f 3 0 c c 2 0 a 1 a 2 e e 2 e 3 A = 0 0 c 0 0 a , σ(a) = 0 0 c 0 0 a e e 6 0 s s f f 6 0 t s e e t 0 0 f f s 0 0 e 0 0 e f 0 0 f , 3.2 The involution ι for C B, P = (p a, q b ) = P B, B Jordan. Let P = (p a, q b ) = (p,..., p; q,..., q), p > q; n = ap + bq. 24

25 a b ap bq Let M = a M(1, 1) M(1, 2), Form M = ap M (1, 1) M (1, 2) b M(2, 1) M(2, 2) as follows: bq M (2, 1) M (2, 2) Replace each entry of M M a+b (K) by small blocks, forming M M n (K). The small blocks of M have the form found in A C B : i. The p p small blocks of M (1, 1) and q q of M (2, 2) are circulant upper triangular (c.u.t.). ii. The q p matrices that comprise the entries of M(2, 1) have the first p q columns zero, followed by a c.u.t. q q subblock C(2, 1) uv, 1 u b, 1 v a. iii. The p q small blocks of M(1, 2) have the last p q rows zero, preceded by a c.u.t. q q subblock B(1, 2) uv, 1 u a, 1 v b. Note: We will use that circulant q q matrices commute. Def. For P = (p a, q b ), We define σ s,p on C B, B = J P. a. apply the involution σ s (a, b) to M, permuting the small blocks. However, in applying σ s (a, b) we must 25

26 b. replace each q p entry M(2, 1) uv = (0, C uv ), 1 u a, 1 v b of M 21 by the p q matrix C uv c. replace each p q entry M(1, 2) uv = of M 21 by the q p matrix (0, B uv ). 0, and B uv 0 1 u b, 1 v a Def. P arbitrary. Define σ = σ s,p : C B C B by defining it for each pair (p, q) of distinct elements of S P. Since the action on the diagonal p p blocks is independent of q, this is consistent. Let K[X P ] the ring of variables, entries of A gen C B ; define ι : K[X P ] K[X P ] by the action of σ s,p on A gen. We also define ι : X 1 (P ) X 1 (P ) by the action of σ on M X1. Thm 3.4 (Involution theorem). i. The involution σ is an antiisomorphism on C B, that restricts to N B and to U B (that is σ : U B U B ). σ(uv ) = σ(v ) σ(u). (3.3) ii. We have for U, V subring K[A gen ] C B : ι(u) = σ(u), and ι(uv ) = ι(u) ι(v ). (3.4) 26

27 iii. Powxe(P ) has the symmetry ι(powxe(p )) = σ(powxe(p )). Ex 3.5. Let P = (3 2, 1 3 ). Then a generic A C B satisfies α 11 a 1 a 2 d d 2 d 3 f 4 f 5 f 6 0 α 11 a 1 0 d d α d α 21 c c 2 α 22 a 3 a 4 f f 2 f 3 A = 0 α 21 c 0 α 22 a , 0 0 α α e e 6 β 11 s s e e 5 β 21 β 22 t 0 0 e 0 0 e 4 β 31 β 32 β 33 π(a) = α β 11 s s 2 11 d, β 21 β 22 t. α 21 α 22 β 31 β 32 β 33 Then σ s,p reflects π(a) about the non-main diagonals. and σ s,p : a 1 a 3, a 2 a 4 ; e f, e i f i, 2 i 6. 27

28 3.3 The vanishing-order matrix Pow(P ); the matrix Powxe(P ) Def. X P = {x ij both A ij 0, A 2 ij = 0, A generic in U B}/mod Hankel relations }. (i.e. We identify equal circulant entries) M X1 (P ) = n n matrix with x ij X P if A generic in U B has entry A ij X P M X1 (P ) ij = 0 otherwise. (3.5) Powxe(P ) = M X1 + (M X1 ) 2 +. Powx(P ) ij = highest degree term of Powxe(P ) ij, Pow(P ) integer matrix, Pow(P ) ij = degree of Powx(P ) ij. Ex 3.6. P = (3), M X1 = 0 a a 0 0 0, Powxe(P ) = 0 a a a

29 Ex 3.7. For P = (3, 1, 1), recall that generic A U(B) and Powxe(P ) satisfy 0 a b f 1 f a 0 0 A =. (3.6) 0 0 e 2 0 c 0 0 e a ce 1 f 1 + a 2 f 1 cf a 0 0 Powxe(P ) =. (3.7) 0 0 ce 1 0 c 0 0 e Here σ : e 1 f 1, e 2 f 2 and ι(powxe(p ) = σ(powxe(p )). Also σ(ce 1 f 1 + a 2 ) = ce 1 f 1 + a 2 - entry fixed by ι; and ι takes ce ( ) ( ) 1 to f 1 cf 1 = σ e 1 ce 1, e 1 29

30 3.4 Constructing Powxe(P ), an example. Ex 3.8. For P = (3 2, 1 3 ). A U B and Pow = Pow(P ): A = 0 a 1 a 2 d d 2 d 3 f 4 f 5 f a 1 0 d d d c c 2 0 a 3 a 4 f f 2 f c 0 0 a e e 6 0 s s e e t 0 0 e 0 0 e , Pow = Here the variables X 1 of M X1 are {c, d, e, f, s, t} and correspond to the entries 1 of Pow(P ). 30

31 We have for P = (3 2, 1 3 ) = (3, 3, 1, 1, 1), Powxe(P ) is 0 cd defst + c 2 d 2 d cd 2 d 2 efst + c 2 d 3 df dfs dfst 0 0 cd 0 d cd d c efst + c 2 d 0 cd defst + c 2 d 2 f fs fst 0 0 c 0 0 cd est 0 0 dest 0 s st 0 0 et 0 0 det 0 0 t 0 0 e 0 0 de Here Q(P ) has two parts (by an R. Basili result, as P = p a, q b, p > q + 1 has r P = 2); the highest nonzero power of a generic A U B is A 6 = d 2 efst E 16, hence Q(P ) = (7, 2). Here Powxe(P ) shows the symmety ι(powxe(p )) = σ(powxe(p ), and is evidently simply constructed from M X1 U B. 31

32 We outline without detail the following result [BaI2, Theorem 3.27]. Thm 3.9 (Algorithm for constructing Powxe(P )).. i. Begin with M X1 (simply defined). ii. The diagonal blocks for each p S P are the same. All other p p blocks are simply constructed from them. iii. Diagonal p p blocks determine terms of q p blocks, q > p, acting via the p q blocks. iv. There is a weaker influence of larger diagonal blocks on smaller ones, when 2q p q. v. Begin with the smallest diagonal blocks, and construct Powxe(P ) in stages, A second algorithm [BaI2, Theorem 3.32] constructs Powxe(P ) by induction on the order of the nonzero entries of each block, using a notion of star product of first lines of Hankel matrices. 32

33 3.5 Pow(P ) and a basis for U B i. 3 Let P = (p r 1 1,..., pr t t ), p 1 > > p t, and let A be a generic element of U B. If the entry A ij 0 and A ij A i 1,j 1 we denote it by x ij, and the set of all such by X P (one variable for each small Hankel diagonal). Let s i = r r i+1. Considering π : C B C B, dim K (U B ) = # X P satisfies # X P = i ( ( )) ri + 1 ir i (r i + 2s i ) r i. (3.8) 2 Let S P = {i r i > 0}, and i S P, j i = r i +max{r i 1, r i+1 } (jump index), s = r i, and recall t =# S P. We denote by X k = {x ij X P A k ij 0 but A k+1 ij = 0} (3.9) Thus, X k comprise the distinct variables from X P corresponding to entries k of Pow(P ). We have [BaI2, Sec. 3.1] # X 1 = s + 2(t 1) # {i j i > r i } (3.10) 3 This section, an algebraic interpretation of some of the results in [BaI2], was inspired by our discussions at the CA meets AC conference January 08 with J. Weyman and T. Koŝir. 33

34 We let B P = I + U B, and filter it by the ideals B P U B U B 2 U B e P 0. Here e P = i(q(p )) 1, i(q(p )) = index of Q(P ), the largest part. We set U 0 B = B P. Denote by E = {e ij, 1 i, j n}, the n 2 -dim vector space. For x ij X P, let v ij E satisfy v ij = e uv where is over {uv A uv = x ij }. Let V k = {v ij, x ij X k }, and V k E their span, V = e P k=1 V k. Thm We have the internal direct sums A. B P = e P k=0 V i = e P k=0 U B k /U B k+1 ; B. for i 0, (U B ) i = k i V k. Proof Outline. We write e ij also for the corresponding element of U B, provided x ij X P. (So U B V ). Let u U k B E have nonzero component on some e ij (with x ij X k ). Then we achieve v ij as a product of k elements v 1 v k, v i V 1. 34

35 Ex (Repeat of Example 2.6). P = (3, 1, 1). 0 x 12 x 13 x 14 x x A =. 0 0 x 43 0 x x Here v 11 = e 12 + e 23, v 12 = e 14, v 13 = e 45, v 14 = e 53 ; v 21 = e 15, v 22 = e 43, v 3 = e 13 U B /U 2 B = V 1 = v 11, v 12, v 13, v 14. U 2 B /U 3 B = V 2 = v 21, v 22 and U 3 B = V 3 = v 3. The action of ι extends to V, and each V i is ι-invariant. Remark. There is symmetry here and for some other (not all) P in the U B -Hilbert functions, when stratified by large matrix blocks, corresponding to 3, (3, 1), 1. Here H UB (V 1 ) = (1, 2, 1), H UB (V 2 ) = (0, 2, 0). Problem. Let A i = generic element of U i B. We have, evidently, rank A i rank A i. Compare these ranks. 35

36 4 What is Q S (P ) maximal nilpotent orbit in π 1 (M S (B))? Nilpotent multi-orbits M S (B) M(B). Definition 4.1. Let P = (p r 1 1,..., pr k k ), p 1 >... > p k. Let r i = POS of partitions of r i. Let S = (S 1,..., S k ), S i r i, 1 i k. Let S(P ) = {S r 1 r k }. M S (B) = nilpotent multi-orbit in M r1 (K) M rk (K) determined by S. Since M S (B) is irreducible and π 1 (M S (B)) is fibred over M S (B) by an affine space isomorphic to the Jacobson radical J of C B, we have π 1 (M S (B)) is irreducible. We denote by Q S (B) the partition giving the Jordan blocks of a generic element of π 1 (M S (B)). Ex 4.2. When S = ((r 1 ),..., (r k )) (each S i a single Jordan block), then M S (B) = M(B), Q S (B) = Q(B). 4 This section was not given at LAW 08, and is from the talk at Dalhousie, Jan

37 Let 0 = S 0 = ((1 r 1),..., (1 r k)) then M S (B) = {(0,..., 0)}, and Q 0 (B) is the maximal partition for an element of J. Observation. When the distinct parts of P differ by two or more, then Q 0 (P ) = P ; otherwise, Q 0 (P ) P. For P = (2, 1 3 ), S = ((1), (1 3 )), then Q 0 (B) = (3, 1, 1) P. Problem: Find Q S (B) for each S. Interpolates between Q(P ), and the generic orbit for A J, the Jacobson radical. Lem 4.3 (Lifting). i. Let σ Gl r1 (K) Gl rk (K) and M, M M(B), and let A C B with π(a) = M. Then there is a unit σ C B such that π(σ (A)) = A. ii. Q S (P ) = P A for A generic in π 1 (J S1,..., J Sk ). That is, in finding Q S (P ) we may assume that π(a) has components each in Jordan block form. 37

38 4.2 The partition Q S (P ) Def: For a fixed P denote by Q(P ) the POS Q(P ) = {Q S (P ) S (P(r 1 ) P(r k ))}, Lem 4.4. : S Q S (P ) is a map of POS: S(P ) Q(P ). For a partition (S 1 = (s 11,..., s 1t ), we let m(s 1 ) = (ms 11,... ms 1t ). Ex 4.5 (Observation). Let P = (m a ) = (m,..., m), and let S 1 be a partition of (a). Then Q S1 (P ) = m(s 1 ). Ex 4.6 (Observation). [Q S (P ) for hooks] Let P = (p, 1 b ) p > 1. Then the map S Q S (P ) : S Q(P ), is an isomorphism of lattices. Q 0 (P ) = P if p 3; Q 0 (P ) = (3, 1 b 1 ) if p = 2. Let S = ((1), R), T P B. Then Q S (P ) is obtained by adding T to Q 0 (P ): add T i 1 to Q 0 (P ) i, i = 1, 2,... until the sum n is attained. Ex P = (2, 1 4 ) (see Ex 3.7B). Q 0 (P ) = (3, 1, 1). S = (2, 2) Q S (P ) = (2, 2) + (3, 1, 1, 1) = ( , ) = (4, 2) 38

39 Ex 4.7. Hooks, p = 2. A. P = (2, 1 3 ); S = 1 3. Q S (P ) S (5) (3) (4, 1) (2, 1) (3, 1, 1) (1, 1, 1) B. P = (2, 1 4 ); S = 1 4. Q S (P ) S (6) (4) (5, 1) (4, 2) (3, 1) (2, 2) (4, 1, 1) (2, 1, 1) (3, 1 3 ) (1 4 ) 39

40 Ex 4.8. Hook: p = 3. P = (3, 1 4 ); S = 1 4. Q S (P ) S (6, 1) (4) (5, 1, 1) (4, 2, 1) (3, 1) (2, 2) (4, 1, 1, 1) (2, 1, 1) (3, 1 4 ) (1 4 ) Ex 4.9. P = (2 2, 1 3 ); S = 2 3. Q S (P ) S (7) (2) (3) (5, 2) (1, 1) (3) (2) (2, 1) (4, 3) (1, 1) (2, 1) (4, 2, 1) (2) (1, 1, 1) (3, 3, 1) (1, 1) (1, 1, 1) S(P ) Q(P ) is not an isomorphism of POS. ((1, 1) (2, 1) and (2) (1, 1, 1) are incomparable in S(P ).) 40

41 4.3 Questions: the involution ι and Q S (P ). a. To what extent is Q S (P ) an invariant of the digraph D(A), or digraph wih involution ι, for A generic in U S (B)? b. What other invariants of P are steps toward Q S (P )? c. Fix P. The condition of A being in π 1 (J r1,..., J rk ) leads to a different digraph-with-involution D than D for A generic in U B. But the lengths of longest paths from i j are unchanged, as the matrix M X1 is in this fibre. Is the S. Poljak calculation of partitions for the generic matrices of digraphs D, D the same? And what is their relation to Q(P )? d. Can the ranks of A k, A generic in U B be concluded from those of certain powers (or powers and sums) of M X1? e. Fix S = (S 1,..., S k ). By regarding the intersection of X 1 (P ) with π 1 (J S1,..., J Sk ), one can construct variables 41

42 X 1 (S) and matrices M X1 (S). Can the ranks of powers of generic elements of the same fibres, be figured from the ranks of powers and sums of M X1 (S)? f. Work in the projectification of C B, N B, and U B. What are the dimensions, closures, and variety structure (CM, irreducible components, types of singularities) of various subvarieties, in particular of the orbits under conjugation by CB. g. What are the intersections of closures of orbits? Relate this problem to analogous problems on the Hilbert scheme. h. Consider a Ellingsrud-Strömme approach to N B : what can one say about homology classes of various subsets, the fixed points of C, torus actions, and the related cellular decomposition? Fix the Hilbert function H. This relates also to the cells G H, Z H (graded, or general quotients of k[x, y] having Hilbert function H). 42

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