When do two nilpotent matrices commute? - Northeastern ITS · When do two nilpotent matrices...

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When do two nilpotent matrices commute? When do two nilpotent matrices commute? Anthony Iarrobino * , Leila Khatami, Bart van Steirteghem, and Rui Zhao Northeastern University Union College Medgar Evers College, CUNY U. Missouri Seminar di Algebra Geometria, Universit` a degli Studi di Genova 25 Giugno, 2014

Transcript of When do two nilpotent matrices commute? - Northeastern ITS · When do two nilpotent matrices...

Page 1: When do two nilpotent matrices commute? - Northeastern ITS · When do two nilpotent matrices commute? Abstract The similarity class of an n by n nilpotent matrix B over a eld k is

When do two nilpotent matrices commute?

When do two nilpotent matrices commute?

Anthony Iarrobino∗, Leila Khatami,Bart van Steirteghem, and Rui Zhao

Northeastern UniversityUnion College

Medgar Evers College, CUNYU. Missouri

Seminar di Algebra Geometria, Universita degliStudi di Genova25 Giugno, 2014

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When do two nilpotent matrices commute?

Abstract

The similarity class of an n by n nilpotent matrix B over a fieldk is given by its Jordan type, the partition P of n that specifiesthe sizes of the Jordan blocks. The variety N (B)parametrizing nilpotent matrices that commute with B isirreducible, so there is a partition Q = Q(P) that is thegeneric Jordan type for matrices A in N (B). The partitionQ(P) has parts that differ pairwise by at least two, and Q(P)is stable: Q(Q(P)) = Q(P).We discuss what is known about the map P to Q(P). A proofof a recursive conjecture by P. Oblak (2008), was recentlyannounced by R. Basili after partial results by P. Oblak,T. Kosir, L. Khatami, and others.What is the set of partitions P having a given partition Q asmaximum commuting orbit? We prove a Table Conjecture ofP. Oblak and R. Zhao when Q has two parts, and generalize itto a Box Conjecture for all stable Q. We also discussequations for the table loci, developed jointly with M. Boij.

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When do two nilpotent matrices commute?

Section 1: The map Q : P → Q(P)

Definition (Nilpotent commutator NB)

V ∼= kn vector space over an infinite field k.A,B ∈ Matn(k) = Homk(V,V);

P ` n partition of n;JP = Jordan block matrix of Jordan type P

CB ⊂ Matn(k) centralizer of B.

NB ⊂ CB : the variety of nilpotent elements of CB .

PA = Jordan type of A.

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When do two nilpotent matrices commute?

Fact: NB is an irreducible variety [Bas1, BI].

Def: Q(P) = PA for A generic in NB ,B = JP .

Problem 1. Given the partition P, determine Q(P)

Fact. Q(P) is Rogers-Ramanujan (RR): the parts of Q(P) differby at least two.

Problem 2. Given the RR partition Q determine Q−1(Q).

Prob. 1: Recursive conjecture of P. Oblak (2008) for Q(P): workof P. Oblak, P. Oblak-T.Kosir, L. Khatami, I-Khatami, R. Basili.

Prob 2: Table conjecture of P. Oblak and R. Zhao (2012,2013) isshown for Q = (u, u − r), r ≥ 2. Box conjecture for Q−1(Q) isopen for Q RR with k > 2 parts..

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When do two nilpotent matrices commute?

Classical problem: but not studied classically. Connected withHilbert scheme work of J. Briancon, M. Granger, R. Basili,V. Baranovsky, A. Premet. See Ngo-Sivic.In 2006, three groups began to work on the P → Q(P) problem,independently

P. Oblak and T. Kosir (Ljubljana)D. Panyushev (Moscow)R. Basili, I.-, and L.Khatami (Perugia, Boston).

Links to work of E. Friedlander, J. Pevtsova, A. Suslin, onrepresentations of Abelian p-groups [FrPS,CFrP]s

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When do two nilpotent matrices commute?

Definition (Almost rectangular)

Let B = J(n), and denote by [n]k = PBk ..

For n = kq, [n]k = (qk) = (q, q, . . . , q).For n = kq + r , 0 < r < k, [n]k =

((dn/ke)r , (bn/kc)k−r

)Here [n]k has k parts that differ at most by 1.We term [n]k almost rectangular (AR).

Ex. n = 5,[5]2 = (3, 2), [5]3 = (2, 2, 1), [5]4 = (2, 1, 1, 1), [5]5 = (1, 1, 1, 1, 1).

Theorem ((R. Basili) Q for rP = 1)

For P = [n]k ,Q(P) = [n] and Q−1([n]) = {[n]k , 1 ≤ k ≤ n}

Example

P = (3, 1) does not commute with (4).

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When do two nilpotent matrices commute?

0 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 10 0 0 0 0

A

0 0 1 0 00 0 0 1 00 0 0 0 10 0 0 0 00 0 0 0 0

A2

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

JP

Figure : A = J[5],A2, and JP where P = [5]2 = (3, 2).

Here A2 is conjugate to JP .

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When do two nilpotent matrices commute?

• • • • •[5]

• • •• •

[5]2

• •• ••

[5]3

• ••••

[5]4

•••••

[5]5

Figure : The AR partitions of 5.

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When do two nilpotent matrices commute?

Example (UB for B = JP ,P = (4))

P = (4),B = JP =

0 1 0 00 0 1 00 0 0 10 0 0 0

. UB : A =

0 xa xb xc0 0 xa xb0 0 0 xa0 0 0 0

A = xaB + xbB2 + xcB3, polynomial in B, soA = uBk , k = 1, 2, 3, 4, u unit in k[B]

PA = [4], or [4]2 = (2, 2) or [4]3 = (2, 1, 1) or [4]4 = (1, 1, 1, 1)

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When do two nilpotent matrices commute?

Theorem (R. Basili [Bas1] )

Q(P) has rP parts, where rP= AR partitions Pi such thatP =

⋃Pi .

Theorem (R. Basili and I.- [BI])

Q(P) = P ⇔ P is RR: the parts of P differ pairwise by at least 2.

Def. We call a P | Q(P) = P “ stable’

also“super-distinct” or “Rogers-Ramanujan” [AlBe, An].

Example

P = (3`, 1`

), Q(P) = (3, 1).

P = ( 5, 4︸︷︷︸, 3, 3, 2︸ ︷︷ ︸, 1`

), Q(P) = (12, 5, 1).

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When do two nilpotent matrices commute?

Poset DP

Rows of vertices: Span the maximal irreducible B - invariantsubspaces of V : each row corresponds to a part of P.Arrows: non-zero elements in A ∈ UB (max subalgebra of NB).

v3•β3

��

v2•β3

��

v1•

v5•

α3

AA

v4•

α3

AA

A =

0 xα3β3 x(α3β3)2 xα3 xα3β3α3

0 0 xα3β3 0 xα3

0 0 0 0 0

0 xβ3 xβ3α3β3 0 xβ3α3

0 0 xβ3 0 0

, v =

v1v2v3v4v5

Figure : Generic element A of UB ,B = JP where P = (3, 2).

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β3

��

β3

��

α3

??

β2

��

α3

??

OO

ε2,1

OO

α2

??

Figure : Diag(DP) for P=(3,2,2,1).

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When do two nilpotent matrices commute?

0 xc3 x(c3)2 xα3 xα3c2 xα3e21 xα3c ′2xα3e21α2

0 0 xc3 0 xα3 0 xα3e21 00 0 0 0 0 0 0 0

0 xe21β3 x43 0 xc2 xe21 xc ′2 xα3e21

0 0 xe21β3 0 0 0 xe21 0

0 0 x63 0 xα2β2 0 xα2β2e21 xα2β2

0 0 xβ3 0 0 0 0 0

0 0 xβ2e21β3 0 xβ2 0 xβ2e21 0

x63 = xα2β2e21α3

Figure : Generic element A of UB for P = (3, 2, 2, 1).

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When do two nilpotent matrices commute?

Relation with Artin algebras

Let PCNn = {pairs A,B of n × n nilp. matrices, [A,B] = 0}.

V. Baranovsky (2001) showed that PCNn is irreducible.When char k = 0 he used a result of J. Briancon (1978) and aproof of M. Granger (1983) that the Hilbert scheme Hilbnk{x , y}parametrizing length-n Artin algebras is irreducible.

R. Basili (2003, char k ≥ n/2) and A. Premet (2003, all infinite k)showed the irreducibility of PCNn directly.This implies the irreducibility of Hilbnk{x , y} for all infinite k.

Pencil Lemma (I.-R. Basili)

Let A = k[A,B] be an Artin algebra with FHS H = H(A) andchar k ≥ n = dimkA. Then PC = H∨, the conjugate of H, forC = A + λB, λ ∈ k generic.

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When do two nilpotent matrices commute?

Theorem (P. Oblak and T.Kosir [KO])

For A ∈ NB generic, the Artin algebra k[A,B] is Gorenstein, so acomplete intersection (CI).

Proof. Uses an involution of the poset DP of NB . See also [BIK,Thm. 2.20].

Corollary (ibid. vith F.H.S. Macaulay [Mac])

Q(P) is stable! (Q(P) is RR: Parts differ pairwise by at least two)

Proof. After Macaulay, if A is CI, the jumps ei = Hi − Hi+1 ofH = H(A) are each less or equal 1, which implies H∨ is RR.

Example

For H = (1, 2, 3, 4, 3, 2, 2, 1),H∨ = (8, 6, 3, 1), which is RR.

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When do two nilpotent matrices commute?

Diagram of the poset DP and maps, P = (4, 2, 2, 1).

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Definition (Poset DP [Obl1, KO, BIK, Kh1])Let P ` n,P = (. . . ini . . .), SP = {i | ni > 0}.The poset DP hasrows of the Ferrers graph of P, each row centered on the y -axis.

There are ni rows of length i :

(u, i , k), 1 ≤ u ≤ i , 1 ≤ k ≤ ni .

Let i−, i+ be the next smaller, next larger elements of SP . Theedges of DP correspond to elementary maps:

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When do two nilpotent matrices commute?

Maps and edges of the diagram DP

(i) βi = βi ,i− : (u, i , ni )→ (u, i−, 1) for u ≤ i−.

(ii) αi = αi−,i : (u, i−, ni−)→ (u + i − i−, i , 1).

(iii) ei ,k : (u, i , k)→ (u, k , k + 1), 1 ≤ ui ≤ i , 1 ≤ k < ni .

(iv) When i is isolated: i − 1 /∈ SP , i + 1 /∈ SP ,

ωi : (u, i , ni )→ (u + 1, i , ni ) for 1 ≤ u < i .

(Each map is 0 on the points of DP not listed)

The diagram of a poset has the covering edges only.The DP is related to a a maximum nilpotent subalgebraUB ⊂ NB ,B = JP : v < v ′ if ∃A ∈ UB | Av ,v ′ 6= 0.

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When do two nilpotent matrices commute?

Def: U-chain in DP determined by an AR P ′ ⊂ P: a chain thatincludes all vertices of DP from an AR subpartition P ′, + two tails.

The first tail descends from the source of DP to the AR chain ofP ′, and the second tail ascends from the AR chain to the sink ofDP .

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When do two nilpotent matrices commute?

Figure : U-chain C4: P = (5, 4, 3, 3, 2, 1) and newU-chain of P ′ = (3, 2, 1).[Source: LK NU GASC talk 2013]

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When do two nilpotent matrices commute?

Oblak Recursive Conjecture

One obtains Q(P) from DP :

(i) Let C be a longest U-chain of DP . Then |C | = q1, thebiggest part of Q(P).

(ii) Remove the vertices of C from DP , giving a partitionP ′ = P − C . If P ′ 6= ∅ then Q(P) = (q1,Q(P ′)) (Go to (i).).

Warning! The poset DP′ is not a subposet of DP .

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When do two nilpotent matrices commute?

Theorem (P. Oblak[Obl1] – Index of Q(P))

The index of Q(P) = is the length of the longest U-chain C of DP .

Theorem (L. Khatami [Kh1] – Ob(P) = λU(DP))

The partition Ob(P) obtained by Oblak recursion is independent ofchoices of AR subpartitions, and Ob(P) = λU(DP), obtained inthe same way as λ(DP) but using U-chains. 1

Work of I-L. Khatami (1/2 Oblak Rec Conj), L. Khatami(smallest part of Q(P)), and R. Basili (Oblak Rec Conj forchar k = 0, 2014) shows the Recursive Conjecture.

1A theory of E.R. Gansner, D. Kleitman, C. Greene, S. Poljak, T. Britz andS. Fomin assigns a partition λ(P), using the lengths of multichains of a poset P

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When do two nilpotent matrices commute?

Section 2: Table conjecture for Q−1(Q).

The set Q−1(Q) is mysterious, even for Q = (u, u − r), r ≥ 2where P → Q(P) is explicit. P. Oblak (2012) [Obl2] and R. Zhao(2013) proposed

Table conjecture for Q−1(Q) (P. Oblak, R. Zhao)

The elements of Q−1(Q),Q = (u, u − r), r ≥ 2 form a(r − 1)× (u − r) table T (Q) such that T (Q)i ,j has i + j parts.

[P. Oblak: # Q−1(Q) = (r − 1)(u − r); R. Zhao: table T (Q)].

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When do two nilpotent matrices commute?

Example (Table T (Q) for Q = (6, 3))

Let Q = (6, 3).

T (Q) =

((6, 3) (6, [3]2) (6, [3]3)

(5, [4]2) (5, [4]3) (5, [4[4)

)=

(A A AB B B

)

• • • • • •• • •

• • • • • •• ••

• • • • • ••••

• • • • •• •• •

• • • • •• •••

• • • • •••••

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When do two nilpotent matrices commute?

Definition (Type A,B,C partitions in Q−1(Q))

Let Q = (u, u − r), r ≥ 2, Q(P) = Q et SP = (a, a− 1, b, b − 1),a > b + 2, or SP = (a, a− 1, a− 2). The largest part u of Q comesfrom a U-row Ca (type A), or Cb (type B) or Ca−1 (type C).

Example

Type A: P = ( 5, 4︸︷︷︸, 2, 1). Type B: P = (5, 4, 2, 2, 2︸ ︷︷ ︸). |C2| = 10

Type C: P = (5, 4, 4, 4, 3, 3︸ ︷︷ ︸, 2), |C4| = 20

Theorem ([Obl2, Z] Special Q−1(u, u − r) )

The table conjecture Q−1 is shown for 2 ≤ r ≤ 4 (P. Oblak); andalso for u >> r – the“normal pattern” case when each A row isfollowed immediately by a B hook (R.Zhao).

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When do two nilpotent matrices commute?

Example (Normal pattern)

The table T (Q) for Q = (6, 3) has “normal pattern”: the first row(6, 3), (6, [3]2), (6, [3]3) is type A, the second(5, [4]2), (5, [4]3), (5, [4]4) is a hook of type B.

• • • • • •• • •

• • • • • •• ••

• • • • • ••••

• • • • •• •• •

• • • • •• •••

• • • • •••••

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When do two nilpotent matrices commute?

Theorem ([IKvSZ] Table T −1(Q))

Let Q = (u, u − r). We can fill the (r − 1)× (u − r) table T (Q)with the partitions from Q−1(Q), arranged in rows of type A andhooks whose partitions have type B or C,B.

T (Q) contains all the set Q−1(Q).

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When do two nilpotent matrices commute?

Example (T (Q) for Q = (8, 3), normal pattern)

Q−1(8, 3) =

(8, 3) (8, [3]2) (8, [3]3)

(5, [6]2) (5, [6]3) (5, [6]4)([8]2, [3]2) ([8]2, [3]3) (5, [6]5)([7]2, [4]3) ([7]2, [4]4) (5, [6]6)

=

A A AB B BA A BB ′ B ′ B

. Note two B hooks.

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When do two nilpotent matrices commute?

Example (T (Q) for Q = (12, 3), First C\A ∪ B case [Z].)

T (Q) 3 [3]2 [3]3

8 (12, 3) (12, [3]2) (12, [3]3)[8]2 ([12]2, 3) [12]2, [3]2) ([12]2, [3]3)[8]3 (5, [10]3) (5, [10]4) (5, [10]5)[8]4 ([12]3, [3]2) ([12]3, [3]3) (5, [10]6)[8]5 (4, [10]4, 1)C ([7]2, [8]5) (5, [10]7)[8]6 ([12]4, [3]3) ([7]2, [8]6) (5, [10]8)[8]7 ([9]3, [6]5) ([7]2, [8]7) (5, [10]9)[8]8 ([9]3, [6]6) ([7]2, [8]8) (5, [10]10)

.

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When do two nilpotent matrices commute?

Idea of proof:

(i) Specify the elements, showing they are in Q−1(Q). X

(ii) Use GF to show #{P | rP = 2,P ` n} is the same as∑|T (Q)| =

∑(r − 1)(u − r), the sum over all RR partitions

Q = (u, u − r), r ≥ 2 de n, (? ) OR

(ii’) Determine all the partitionsl P of type C having Q(P) = Qand show that are in T (Q). X

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When do two nilpotent matrices commute?

Table Loci Theorem

(In process: with M.Boij, T.Kosir, K.Sivic, P. Oblak, L. Khatami,Bart van Steirteghem).

(i) The locus Zij in P(UB) of all matrices A of Jordan typeTij(Q),Q = (u, u − r), 1 ≤ i ≤ r − 1, 1 ≤ j ≤ u − r is acomplete intersection of codimension i + j − 2 in P(UB).

(ii) The locus Zij is given by equations of whichmin{i + j − 2, r − 2} are linear, and k = max{i + j − r , 0} arequadratic (k − 1)-th polarizations of a 2× 2 determinantunique to the A-row or B hook.

Case Q = (6, 3).

T =

((6, 3) (6, [3]2) (6, [3]3)

(5, [4]2) (5, [4]3) (5, [4]4)

)E =

(− t t,M0

a a,M a,M,N

),.

M0 =

(a gg ′ u

), M =

(b gg ′ t

), N =

(c hg ′ t

)+

(b gh′ u

).

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When do two nilpotent matrices commute?

Coordinates in NB ,B = JQ ,Q = (6, 3)

A =

0 a b c d e g h i0 0 a b c d 0 g h0 0 0 a b c 0 0 g0 0 0 0 a b 0 0 00 0 0 0 0 a 0 0 00 0 0 0 0 0 0 0 00 0 0 g ′ h′ i ′ 0 t u0 0 0 0 g ′ h′ 0 0 t0 0 0 0 0 g ′ 0 0 0

,

B :a = t = 1

{b, . . . e, g . . . i , g ′ . . . i ′, u} = 0..

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Example (Equations for table loci: T (Q),Q = (6, 3))

T =

((6, 3) (6, [3]2) (6, [3]3)

(5, [4]2) (5, [4]3) (5, [4]4)

)

E =

(− t t,M0

a a,M a,M, |N| = 0

),M =

(b gg ′ t

),N =

c hb gh′ ug ′ t

.

M0 =

(a gg ′ u

).

|N| = 0 : ct − g ′h + bu − h′g = 0.

((c hg ′ t

)+

(b gh′ u

)= 0

)

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When do two nilpotent matrices commute?

Section 3: Box Conjecture

.

Definition (Key SQ of a stable Q)

Let Q = (q1, q2, . . . , qk), qi ≥ qi+1 + 2, 1 ≤ i < k be stable. Thekey SQ = (q1 − q2 − 1, q2 − q3 − 1, . . . , qk−1 − qk − 1, qk).

Example

For Q = (u, u − r) the key is SQ = (r − 1, u − r).For Q = (12, 6, 2) the key is SQ = (5, 3, 2)

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When do two nilpotent matrices commute?

Box conjecture for Q−1(Q)

Let Q = (q1, . . . , qk) be stable of key SQ . Then

(i) The partitions Q−1(Q) form a k-box T (Q) such thatT (Q)I , I = (i1, . . . , ik) has |I | parts.

(ii) The codimension of the similarity orbit of T (Q)I in NQ is|I | − k.

(iii) The locus Z(PI ) is an irreducible complete intersection definedby linear and quadratic equations in the variables ofUB ,B = JQ .

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When do two nilpotent matrices commute?

Example (SQ = (2, 2, 2))

Take Q = (8, 5, 2) so SQ = (2, 2, 2).The two floors of T (Q) are(

(8, 5, 2) (8, 5, 12)(8, 4, 2, 1) (8, 4, 13)

),

((7, 4, 22) (7, 4, 2, 12)(7, 32, 12) (7, 4, 14)

).

The corresponding floors of DH(Q) = θ(T (Q)) are((6, 5, 4) (5, 4, 32)(5, 42, 2) (4, 33, 2)

),

((52, 4, 1) (42, 32, 1)(43, 2, 1) (34, 2, 1)

).

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Question: Can we explain these results? Not yet!

Lie algebra perspective:

The columns of D(P) are weight spaces for the sl2 triple of B. Butthe Sn irreps for P ∈ T (Q) and θ(P) ∈ DH(Q) have different VSdimensions.

Map to the Hilbert scheme:

Let B = JQ . The mapπ : NB → Hilbnk[x , y ]: A→ k[A,B]

defines an image, whose fixed points under a C∗ action correspondto the monomial ideals of T (Q), so to homology classes onπ(NB). Will this explain the codimensions in T (Q)?

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Combinatorial questions arising from P → Q(P).

(a) Poset D(P): Is λ(DP) = λU(DP)?

(b) Explain the map θ−1 : DH(Q)→ T (Q) combinatorially.

(c) Verify #{P` n with p parts and rP = k} is the expected sum.

(d) An a-cluster is a partition P = (p1 ≥ . . . ≥ pt) withp1 − pt ≤ a.ra,P = min{ # a-clusters needed to cover P}.Va,k(n) = {P ` n | ra,P = k}.Determine |Va,k(n)|.(e) Consider other posets P with multiplicities, and a linear actionB → on vertices(P). Consider A ∈ I(P) commuting with B.Is λ(P) = λB(P)?

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Acknowledgment

We appreciate discussions with and helpful comments byDon King, Alfred Noel, George McNinch, and a conversation ofRui and Tony with Barry Mazur. We are grateful for the insights ofP. Oblak, T. Kosir and others who contributed questions andresults that have been important to our work. We appreciate useof notes of Rick Porter on LaTeX, xy-pic, and his advice.

Thank you for your attention and questions!

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Appendix: Q(P) and its smallest part (L.Khatami)

β3

��

β3

��

α3

??

β2

��

α3

??

e21

OO

e2,1

OO

α2

??

Figure : Diagram of the poset DP : P = (3, 2, 2, 1).

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Def. (U-chain)

A U-chain Ci in DP is the saturated (maximal) chain through theunion of three subsets of vertices:

(i) All rows of length i , i − 1, corresponding to an ARsubpartition of P.

(ii) A descending chain from the source – the top left vertex ofDP – to the vertex at the start of the lowest length-i row.

(iii) An ascending chain from the vertex at the end of the highestlength-i row to the sink - the top right vertex of DP .

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Figure : U-chain C4 for P = (5, 4, 3, 3, 2, 1) and newU-chain for P ′ = (3, 2, 1).[Source: LK NU GASC talk 2013]

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Oblak Recursive Conjecture

We obtain Q(P) as follows from DP :

(i) Choose a maximum length U-chain in DP . Its length is q1,the largest part of Q(P).

(ii) Remove the vertices in the chain from DP , obtaining a smallerpartition P ′. If P ′ 6= ∅ then Q(P) = (q1,Q(P ′)) (go to (i).).

Warning. The poset DP′ in the Oblak recursion is not in general asubposet of DP .

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Figure : U-chain for P = (5, 4, 3, 3, 2, 1) and newU-chain for P ′ = (3, 2, 1). So Q(P) = (12, 5, 1).

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Theorem (P. Oblak[Obl1] – Index of Q(P))

The index (largest part) of Q(P) is the length of the longestU-chain in DP .

Theorem (L. Khatami [Kh1] – Ob(P) = λU(DP))

The partition Ob(P) obtained by the Oblak recursive process isindependent of the choices of AR subpartitions; andOb(P) = λU(DP), obtained as λ(DP) below by restricting to setsof U-chains.

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Definition

P ≥ P ′ in the orbit closure (Bruhat) order if∀i∑i

u=1 pu ≥∑i

u=1 p′u.

Theorem (I,L.Khatami [IKh])

Q(P) ≥ Ob(P).

Proof idea. For each maximal-length set of s U-chains, we specifyan A ∈ NB such that dimk k[A] ◦ {v1, . . . , vs} where the vi areinitial elements, agrees with the sum of the first s parts of Ob(P).This involves an analysis of the sets of chains from the vi to all thevertices covered by the s U-chains.

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Def. (C. Greene et al, see[BrFo])

Let D be a poset without loops. Define ci = max# verticescovered by i chains. Set

λ(D) = (c1, c2 − c1, c3 − c2, . . .).

Theorem (C. Greene, S. Poljak, E.R. Gansner, see [BrFo])

Let D be any finite poset without loops, and let A be a genericnilpotent matrix in the incidence algebra I(DP). Then the Jordantype PA = λ(D).

Definition ([Kh1])

λU(DP) is obtained by replacing arbitrary chains ci in thedefinition of λ(DP) by U-chains.

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Example (Case rP = 1, dh(P) has 1× 1 Durfee square.)

Let n = 5,Q = (5).T (Q) =

([5], [5]2, [5]3, [5]4, [5]5

)DH(Q) =

((5), (4, 1), (3, 12), (2, 14), (15)

)(single diagonal hook).

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Table Loci Theorem

(In process: with M.Boij, T.Kosir, K.Sivic, P. Oblak, L. Khatami,Bart van Steirteghem).

(i) The locus Zij in P(UB) of all matrices A of Jordan typeTij(Q),Q = (u, u − r), 1 ≤ i ≤ r − 1, 1 ≤ j ≤ u − r is acomplete intersection of codimension i + j − 2 in P(UB).

(ii) The locus Zij is given by equations of whichmin{i + j − 2, r − 2} are linear, and k = max{i + j − r , 0} arequadratic (k − 1)-th polarizations of a 2× 2 determinantunique to the A-row or B hook.

Case Q = (6, 3).

T =

((6, 3) (6, [3]2) (6, [3]3)

(5, [4]2) (5, [4]3) (5, [4]4)

)E =

(− t t,M0

a a,M a,M,N

),.

M0 =

(a gg ′ u

), M =

(b gg ′ t

), N =

(c hg ′ t

)+

(b gh′ u

).

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Definition (Key SQ of a stable Q)

Let Q = (q1, q2, . . . , qk), qi ≥ qi+1 + 2, 1 ≤ i < k be stable. Thekey SQ = (q1 − q2 − 1, q2 − q3 − 1, . . . , qk−1 − qk − 1, qk).

Example

For Q = (u, u − r) the key is SQ = (r − 1, u − r).For Q = (12, 6, 2) the key is SQ = (5, 3, 2)

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Box conjecture for Q−1(Q)

Let Q = (q1, . . . , qk) be stable of key SQ . Then

(i) The partitions Q−1(Q) form a k-box T (Q) such thatT (Q)I , I = (i1, . . . , ik) has |I | parts.

(ii) The codimension of the similarity orbit of T (Q)I in NQ is|I | − k.

Corollary of the box conjecture

For Q stable, there is an isomorphism θ : Q−1(Q)→ DH(Q), thatpreserves numbers of parts.

Problems Find θ explicitly.Give the table T (Q). (Find “hooks” for k ≥ 3.)

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Department of Mathematics, Northeastern University,Boston MA 02115, USAE-mail address: [email protected]

Department of Mathematics, Union College, Schenectady,NY 12308, USAE-mail address: [email protected]

Department of Mathematics, Medgar Evers College, CityUniversity of New York, Brooklyn, NY 11225, USAE-mail address: [email protected]

Mathematics Department, University of Missouri,Columbia, MO, 65211, USAE-mail address: [email protected]