Parallel transport in the Kazhdan-Lusztig W -graph and Greens 0 1 - PowerPoint PPT Presentation

Parallel transport in the Kazhdan-Lusztig W -graph and Green’s 0 − 1 conjecture in Lie type B Michael Chmutov University of Minnesota September 20, 2014 AMS Meeting #1102 Eau Claire, WI Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

The Hecke Algebra and Kazhdan-Lusztig polynomials ( W , S ) – Coxeter system, ground ring: Z [ q ± 1 / 2 ] � T s , s ∈ S � T s T t T s · · · = T t T s T t . . . H = ( T s + 1)( T s − q ) = 0 Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

The Hecke Algebra and Kazhdan-Lusztig polynomials ( W , S ) – Coxeter system, ground ring: Z [ q ± 1 / 2 ] � T s , s ∈ S � T s T t T s · · · = T t T s T t . . . H = ( T s + 1)( T s − q ) = 0 Standard basis { T w } w ∈ W ; Kazhdan-Lusztig basis { C w } w ∈ W Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

The Hecke Algebra and Kazhdan-Lusztig polynomials ( W , S ) – Coxeter system, ground ring: Z [ q ± 1 / 2 ] � T s , s ∈ S � T s T t T s · · · = T t T s T t . . . H = ( T s + 1)( T s − q ) = 0 Standard basis { T w } w ∈ W ; Kazhdan-Lusztig basis { C w } w ∈ W Transition matrix entries, up to a power of q , are Kazhdan-Lusztig polynomials P v , w ( q ) Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

The Hecke Algebra and Kazhdan-Lusztig polynomials ( W , S ) – Coxeter system, ground ring: Z [ q ± 1 / 2 ] � T s , s ∈ S � T s T t T s · · · = T t T s T t . . . H = ( T s + 1)( T s − q ) = 0 Standard basis { T w } w ∈ W ; Kazhdan-Lusztig basis { C w } w ∈ W Transition matrix entries, up to a power of q , are Kazhdan-Lusztig polynomials P v , w ( q ) � � l ( w ) − l ( v ) − 1 deg ( P v , w ) � l ( w ) − l ( v ) − 1 ; µ ( v , w ) = P v , w (symmetrized) q 2 2 Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 Vertices: elements of W s s s s 2 1 1 2 s 1 s 2 1 Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 Vertices: elements of W s s s s Edges: weighted by µ values 2 1 1 2 s 1 s 2 1 Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 12 Vertices: elements of W s s s s Edges: weighted by µ values 2 1 1 2 2 1 τ -labels: left descent sets 1 2 s 1 s 2 1 Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 12 Vertices: elements of W s s s s Edges: weighted by µ values 2 1 1 2 2 1 τ -labels: left descent sets Directions: based on τ -label 1 2 s 1 s 2 containments (new weights m ( x , w )) 1 Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 12 Vertices: elements of W s s arc s s Edges: weighted by µ values 2 1 1 2 2 1 τ -labels: left descent sets simple edge Directions: based on τ -label 1 2 s 1 s 2 containments (new weights m ( x , w )) 1 Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 12 Vertices: elements of W s s arc s s Edges: weighted by µ values 2 1 1 2 2 1 τ -labels: left descent sets simple edge Directions: based on τ -label 1 2 s 1 s 2 containments (new weights m ( x , w )) 1 0 − 1 Conjecture In type A , the edge weights are always 0 or 1. Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 12 Vertices: elements of W s s arc s s Edges: weighted by µ values 2 1 1 2 2 1 τ -labels: left descent sets simple edge Directions: based on τ -label 1 2 s 1 s 2 containments (new weights m ( x , w )) 1 0 − 1 Conjecture (disproved; McLarnan and Warrington, 2003) In type A , the edge weights are always 0 or 1. Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 12 Vertices: elements of W s s arc s s Edges: weighted by µ values 2 1 1 2 2 1 τ -labels: left descent sets simple edge Directions: based on τ -label 1 2 s 1 s 2 containments (new weights m ( x , w )) 1 0 − 1 Conjecture (disproved; McLarnan and Warrington, 2003) In type A , the edge weights are always 0 or 1. Green’s 0 − 1 Conj. The edge weights m ( x , w ) are 0 or 1 when x is fully commutative. Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Kazhdan-Lusztig W -graph and 0-1 conjectures s s s 1 2 1 12 Vertices: elements of W s s arc s s Edges: weighted by µ values 2 1 1 2 2 1 τ -labels: left descent sets simple edge Directions: based on τ -label 1 2 s 1 s 2 containments (new weights m ( x , w )) 1 0 − 1 Conjecture (disproved; McLarnan and Warrington, 2003) In type A , the edge weights are always 0 or 1. Green’s 0 − 1 Conj. A , � A (Green, 2009); D (Gern, 2013); B (C., 2014) The edge weights m ( x , w ) are 0 or 1 when x is fully commutative. Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Fully commutative elements Any two reduced expressions of w are connected by braid moves. Weak Order Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Fully commutative elements Any two reduced expressions of w are connected by braid moves. Defintion w is fully commutative if only length 2 braid relations necessary. Weak Order Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Fully commutative elements Any two reduced expressions of w are connected by braid moves. Defintion w is fully commutative if only length 2 braid relations necessary. Weak Order Index basis of generalized Temperley-Lieb algebra, Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Fully commutative elements Any two reduced expressions of w are connected by braid moves. Defintion w is fully commutative if only length 2 braid relations necessary. Weak Order Index basis of generalized Temperley-Lieb algebra, In type A , 321-avoiding, Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Fully commutative elements Any two reduced expressions of w are connected by braid moves. Defintion w is fully commutative if only length 2 braid relations necessary. Weak Order Index basis of generalized Temperley-Lieb algebra, In type A , 321-avoiding, In type B , avoids 12, 321, 321, 321,321, 312, 312, 213,231, 231, 132, Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Fully commutative elements Any two reduced expressions of w are connected by braid moves. Defintion w is fully commutative if only length 2 braid relations necessary. Weak Order Index basis of generalized Temperley-Lieb algebra, In type A , 321-avoiding, In type B , avoids 12, 321, 321, 321,321, 312, 312, 213,231, 231, 132, τ -labels do not contain adjacent simple reflections. Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Simple edges are simple Properties of simple edges: Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Simple edges are simple Properties of simple edges: Weight 1 both ways, Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Simple edges are simple Properties of simple edges: Weight 1 both ways, Preserve full commutativity, Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Simple edges are simple Properties of simple edges: Weight 1 both ways, Preserve full commutativity, Any simple edge activates a pair ( s , t ) of simple reflections, s t Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Simple edges are simple Properties of simple edges: Weight 1 both ways, Preserve full commutativity, Any simple edge activates a pair ( s , t ) of simple reflections, s t If order ( st ) = 3, graph restricted to parabolic � s , t � is in (a). ( a ) ( b ) 2 st 12 1 1 12 st 1 t 2 s 2 s 2 1 t s 1 t Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

Simple edges are simple Properties of simple edges: Weight 1 both ways, Preserve full commutativity, Any simple edge activates a pair ( s , t ) of simple reflections, s t If order ( st ) = 3, graph restricted to parabolic � s , t � is in (a). If order ( s 1 s 2 ) = 4, graph restricted to parabolic � s 1 , s 2 � is in (b). ( a ) ( b ) 2 st 12 1 1 12 st 1 t 2 s 2 s 2 1 t s 1 t Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B