October 14, 2016 1 / 31 Tetrahedral Geometry and Topology Seminar - PowerPoint PPT Presentation

October 14, 2016 1 / 31

Tetrahedral Geometry and Topology Seminar Deformation Theory Seminar October 14, 2016 Seaweed and poset algebras synergies and cohomology Part I Seaweeds by Vincent E. Coll, Jr.* A report on joint work with Matt Hyatt** and Colton Magnant*** *Lehigh University **Pace University ***Georgia Southern University October 14, 2016 2 / 31

Objects and definitions OBJECTS Frobenius seaweed subalgebras of simple Lie algebras Simple A n = sl ( n ), B n = so (2 n + 1), C n = sp (2 n ), D n = so (2 n ) E 6 , E 7 , E 8 , F 4 , G 2 Frobenius B F [ x , y ] = F [ x , y ] ind g = min F ∈ g ∗ dim ker B F . Seaweeds g simple p , p ′ parabolic subalgebras with p + p ′ = g p ∩ p ′ is a seaweed October 14, 2016 3 / 31

Type-A seaweeds Special linear Lie algebra sl ( n ) = { A ∈ gl( n ) | tr( A ) = 0 } Type-A seaweed 2 2 * * * * * * * * * * * 1 * * * * * * * 4 1 * * * 4 2 * * * * * * * * 2 * * * * * * * * * * * * * * * * 1 * * * * * * 1 p ∩ p ′ = p A 4 | 1 p ′ p 5 2 | 1 | 2 October 14, 2016 4 / 31

Index computation – Meanders 2 | 2 | 3 Type p A 7 5 | 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 v 2 v 4 v 6 v 7 v 3 v 5 v 6 v 1 v 2 v 4 v 7 v 3 v 5 v 6 v 1 v 2 v 4 v 7 v 3 v 5 v 6 v 1 v 2 v 4 v 7 v 3 v 5 Theorem (Dergachev and Kirillov, 2000) For a seaweed g ⊆ sl ( n ) , ind g = 2 C + P − 1 October 14, 2016 5 / 31

Not so easy in practice 5 | 7 | 4 | 10 Type: 8 | 6 | 6 | 6 October 14, 2016 6 / 31

Not so easy in practice 5 | 7 | 4 | 10 Type: 8 | 6 | 6 | 6 Index: 2 October 14, 2016 6 / 31

Index formulas Theorem (Elashvilli (1990), C., Magnant, and Giaquinto (2010)) If n = a + b with ( a , b ) = 1 , then a | b is Frobenius. n Theorem (C., Hyatt, Magnant, and Wang (2015)) If n = a + b + c with ( a + b , b + c ) = 1 , then a | b | c is Frobenius. n Theorem (Karnauhova and Liebscher (2015)) If m ≥ 4 , then ∄ homogeneous f 1 , f 2 ∈ Z [ x 1 , . . . , x m ] such that inda 1 | a 2 | · · · | a m = gcd( f 1 ( a 1 , . . . , a m ) , f 2 ( a 1 , . . . , a m )) . n October 14, 2016 7 / 31

The signature of a meander a 1 | a 2 | · · · | a m − → M b 1 | b 2 | · · · | b t Condition Move a 1 = b 1 Component removal a 1 = 2 b 1 Block removal b 1 < a 1 < 2 b 1 Rotation a 1 > 2 b 1 Pure a 1 < b 1 Flip October 14, 2016 8 / 31

6 | 1 Detailed signature of p A 7 2 | 3 | 2 October 14, 2016 9 / 31

Type-A Frobenius functional, F A * * * * * * * * 3 | 2 p A * * * * 5 5 1 2 3 4 5 * * * F A = e ∗ 1 , 5 + e ∗ 2 , 4 + e ∗ 3 , 1 + e ∗ 5 , 4 is regular October 14, 2016 10 / 31

Type-A principal element, ˆ F A 1 | 4 p A 1 2 3 4 5 5 2 | 3 Pick an endpoint of the path, say vertex 4 Follow the path from 1 to 4, counting the arrows ( measure ) The measure is − 2, this is the first diagonal entry of D . Repeat for each vertex: D = diag ( − 2 , − 3 , − 1 , 0 , − 2) Normalize: ˆ F A = D + 8 / 5 = diag ( − 2 / 5 , − 7 / 5 , 3 / 5 , 8 / 5 , − 2 / 5) October 14, 2016 11 / 31

Eigenvalues of ad ˆ F A 1 2 3 4 5 0 1 0 2 0 -1 1 3 1 0 2 1 -1 -2 Pick a pair of vertices (4,2) Measure is 3 This is an eigenvalue: ad ˆ F A ( e 4 , 2 ) = 3 e 4 , 2 Eigenvalues − 2 − 1 0 1 2 3 Dimensions 1 2 4 4 2 1 October 14, 2016 12 / 31

Type-A unbroken spectrum result Confirming a claim of Gerstenhaber and Giaquinto – and a bit more. Theorem (C., Hyatt, Magnant (2016)) For a seaweed subalgebra of sl ( n ) , the spectrum of ad ˆ F is an unbroken sequence of integers. Moreover, the multiplicities form a symmetric distribution. Proof uses the Signature. October 14, 2016 13 / 31

Type-C seaweeds Symplectic Lie algebra �� A � � B : B = � B , C = � sp (2 n ) = , C − � C A A , B , and C are n × n matrices. � A is the transpose of A with respect to the anitdiagonal. Type-C seaweeds a 1 | · · · | a m p C n b 1 | · · · | b t where the a 1 | · · · | a m and b 1 | · · · | b t are partial compositions of n , i.e. � a i ≤ n and � b i ≤ n . October 14, 2016 14 / 31

Type-C (symplectic) seaweeds, cont... 2 | 1 | 1 | 6 p C 11 2 | 2 | 1 | 2 2 * * 2 2 * * 1 * * 1 1 * 2 * * * * * * * 6 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * October 14, 2016 15 / 31

...and the associated meander of Blocks V i = { vertices : a 1 + a 2 + · · · + a i − 1 < vertex label < a 1 + a 2 + · · · a i + 1 Tail Let r = n − � a i and T n ( a ) = { n − r + 1 , n − r + 2 , . . . , n } T = ( T n ( a ) ∪ T n ( b )) \ ( T n ( a ) ∩ T n ( b )) 1 2 3 4 5 6 7 8 9 10 11 Top blocks: V 1 = { 1 , 2 } , V 2 = { 3 } , V 3 = { 4 } , V 4 = { 5 , 6 , 7 , 8 , 9 , 10 } Bottom blocks: V 1 = { 1 , 2 } , V 2 = { 3 , 4 } , V 3 = { 5 } , V 4 = { 6 , 7 } Tail: T = { 8 , 9 , 10 } October 14, 2016 16 / 31

2 | 1 | 1 | 6 Index computation for p C 11 2 | 2 | 1 | 2 Theorem (C., Hyatt., and Magnant (2016)) For a seaweed g ⊆ sp (2 n ) , ind g = 2 C + ˜ P ˜ P is number of paths with 0 or 2 edges in the tail. 1 2 3 4 5 6 7 8 9 10 11 component # of tail vertices cycle? contribution to index G [ { 1 , 2 } ] 0 yes 2 G [ { 3 , 4 } ] 0 no 1 G [ {{ 11 } ] 0 no 1 G [ { 6 , 7 , 8 , 9 } ] 2 no 1 G [ { 5 , 10 } ] 1 no 0 October 14, 2016 17 / 31

Index of symplectic seaweeds Index Formulas Theorem (C., Hyatt, and Magnant) n ind p C a | b = 0 iff one of the following holds: n a + b = n − 1 and gcd( a + b , b + 1) = 1 a + b = n − 2 and gcd( a + b , b + 2) = 1 a + b = n − 3 and gcd( a + b , b + 3) = 2 with n , a , and b all odd. October 14, 2016 18 / 31

Corollary – Frobenius Type-C meanders are ... ...a certain kind of forest... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 October 14, 2016 19 / 31

Type-C Frobenius functional, F C Panyushev-Yakimova meander 1 2 3 4 5 6 7 8 9 10 mirror Theorem (C., Hyatt, and Magnant (2015)) � e ∗ F C = i , j , such that i ≤ n or j ≤ n, is Frobenius. ( i , j ) Example F C = e ∗ 1 , 2 + e ∗ 3 , 4 + e ∗ 3 , 1 + e ∗ 6 , 5 + e ∗ 7 , 4 is Frobenius. October 14, 2016 20 / 31

Type-C principal element, ˆ F C Principal Graph m 1 2 3 4 5 6 7 8 9 10 Edges incident with m have measure 1 / 2, all other edges have measure 1. Measure from each vertex to m. � � − 1 2 , − 3 2 , 1 2 , − 1 2 , − 1 2 , 1 2 , 1 2 , − 1 2 , 3 2 , 1 ˆ F C = diag 2 October 14, 2016 21 / 31

Eigenvalues of ˆ F C 1 2 3 4 5 6 7 8 9 10 0 1 -1 0 1 2 0 1 0 0 0 1 1 1 October 14, 2016 22 / 31

Type-C unbroken spectrum result Eigenvalues − 1 0 1 2 Dimensions 1 6 6 1 October 14, 2016 23 / 31

Type-C unbroken spectrum result Eigenvalues − 1 0 1 2 Dimensions 1 6 6 1 Theorem (C., Hyatt., and Magnant (2016)) The spectrum of a principal element of Frobenius symplectic seaweed is an unbroken sequence of integers. Moreover the multiplicities form a symmetric sequence. Proof uses two ingredients Type-A unbroken result Adaptation of the signature October 14, 2016 23 / 31

Type-B seaweeds Special orthogonal Lie algebra so (2 n + 1) = { A ∈ gl( n ) | A = − � A } , a 1 | · · · | a m p B n b 1 | · · · | b t where the a 1 | · · · | a m and b 1 | · · · | b t are partial compositions of n , i.e. � a i ≤ n and � b i ≤ n . October 14, 2016 24 / 31

Type-B seaweeds cont... 1 | 1 p B 5 4 | 1 | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 October 14, 2016 25 / 31

1 | 1 p C 5 4 | 1 | 1 F C = e ∗ 1 , 4 + e ∗ 2 , 3 + e ∗ 10 , 3 + e ∗ 9 , 4 + e ∗ 8 , 5 + e ∗ 7 , 6 Read DIRECTLY from the meander 1 2 3 4 5 6 7 8 9 10 11 12 mirror October 14, 2016 26 / 31

1 | 1 p B 5 4 | 1 | 1 0 0 0 0 0 0 0 F B = e ∗ 1 , 4 + e ∗ 2 , 3 + e ∗ 10 , 3 + e ∗ 9 , 4 + e ∗ 8 , 5 + e ∗ 7 , 3 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 mirror October 14, 2016 27 / 31

(It seems that) A stochasitic process is present 40 40 30 30 multiplicity multiplicity 20 20 10 10 0 0 − 3 − 2 − 1 0 1 2 3 4 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 eigenvalue eigenvalue 4 | 10 | 3 4 | 13 Distribution of eigenvalues of p A 6 | 4 | 7 (left) and p A (right). 17 17 17 October 14, 2016 28 / 31

More precisely... What seems to be true 1. Distribution is unimodular 2. Let λ ∈ Z + F λ be a Frobenius seaweed with spectrum { 1 − λ, ..., 0 , ...., λ } d i ( F λ ) = dim of the λ -eigenspace { F λ } λ = ∞ λ =1 be a sequence of such Frobenius seaweeds { X λ } λ = ∞ λ =1 be a sequence of random variables P ( X λ = i ) = d i ( F λ ) dimF λ X λ − → Normal (in distribution). October 14, 2016 29 / 31

Lie poset algebras The “Stargate” poset 0 * * * 4 0 * * * 0 * * 3 0 0 * 0 0 1 2 Rank g ( P , C ) = 3 P = { 1 , 2 , 3 , 4 } g ( P , C ) = 8 1 , 2 � 3 � 4 dim F = e ∗ 1 , 4 + e ∗ 2 , 4 + e ∗ 2 , 3 � 1 � ˆ 2 , 1 2 , − 1 2 , − 1 F = diag 2 Spec ad ˆ F = { 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 } . October 14, 2016 30 / 31

But,... g ( P , C ) is NOT a seaweed! The only seaweed subalgebra of sl ( 4 ) with Rank 3 and dimension 8 is 2 | 2 p A 4 1 | 3 It’s spectrum is {− 1 , 0 , 0 , 0 , 1 , 1 , 1 , 2 } October 14, 2016 31 / 31