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Parallel transport in the Kazhdan-Lusztig W -graph and Greens 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

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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

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The Hecke Algebra and Kazhdan-Lusztig polynomials

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Michael Chmutov

Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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The Hecke Algebra and Kazhdan-Lusztig polynomials

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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The Hecke Algebra and Kazhdan-Lusztig polynomials

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries, up to a power of q, are Kazhdan-Lusztig polynomials Pv,w(q)

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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The Hecke Algebra and Kazhdan-Lusztig polynomials

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries, up to a power of q, are Kazhdan-Lusztig polynomials Pv,w(q) deg(Pv,w) l(w)−l(v)−1

2

; µ(v, w) =

  • q

l(w)−l(v)−1 2

  • Pv,w (symmetrized)

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W 1

1 2 1 2

s s

1

s s

1 2

s1 s2 s s s

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values τ-labels: left descent sets s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1 2 1 2 12 1

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values τ-labels: left descent sets Directions: based on τ-label containments (new weights m(x, w)) s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1 2 1 2 12 1

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values τ-labels: left descent sets Directions: based on τ-label containments (new weights m(x, w)) s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1 arc simple edge 1 12 2 1 2

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values τ-labels: left descent sets Directions: based on τ-label containments (new weights m(x, w)) s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1 arc simple edge 1 12 2 1 2 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

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values τ-labels: left descent sets Directions: based on τ-label containments (new weights m(x, w)) s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1 arc simple edge 1 12 2 1 2 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

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values τ-labels: left descent sets Directions: based on τ-label containments (new weights m(x, w)) s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1 arc simple edge 1 12 2 1 2 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

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Kazhdan-Lusztig W -graph and 0-1 conjectures

Vertices: elements of W Edges: weighted by µ values τ-labels: left descent sets Directions: based on τ-label containments (new weights m(x, w)) s s s

1 2 1 2

s s

1

s s

1 2

s1 s2 1 arc simple edge 1 12 2 1 2 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

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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

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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

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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. Index basis of generalized Temperley-Lieb algebra,

Weak Order

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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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. Index basis of generalized Temperley-Lieb algebra, In type A, 321-avoiding,

Weak Order

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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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. 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,

Weak Order

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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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. 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.

Weak Order

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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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

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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

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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

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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,

t s

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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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,

t s

If order(st) = 3, graph restricted to parabolic s, t is in (a). (a) (b)

s st s t s t t st 12 12 2 2 2 1 1 1 1 1 2

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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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,

t s

If order(st) = 3, graph restricted to parabolic s, t is in (a). If order(s1s2) = 4, graph restricted to parabolic s1, s2 is in (b). (a) (b)

s st s t s t t st 12 12 2 2 2 1 1 1 1 1 2

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Parallel transport in simply laced types

For s, t with order(st) = 3 and

w x’ w’ x s s t t

we have µ(x, w) = µ(x′, w ′).

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Example: A5

Length 1 edges (in Bruhat order) have weights 0 or 1.

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Example: A5

24 134 23 13 125 124 235 14 24 34 35 245 145 135 25 135

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Example: A5

24 134 23 13 125 124 235 14 24 34 35 245 145 135 25 135

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Example: A5

24 134 23 13 125 124 235 14 24 34 35 245 145 135 25 135

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Example: A5

23 134 13 125 124 235 14 24 34 35 245 145 135 25 24 135

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Example: A5

24 134 23 13 125 124 235 14 24 34 35 245 145 135 25 135

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Parallel transport in type B

n

2 3 1 n B :

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Parallel transport in type B

n

2 3 1 n B :

Additional relations for the double bond: a+b b a a b 1 2 1 1 2 1 2 a a a 1 2 1 2 1 a

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Proof ideas

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Proof ideas

Theorem (Shi, 2003); types A,B,. . . From non-f.c. element can reach element with adjacent entries in τ-invariant via simple edges (move down).

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Proof ideas

Theorem (Shi, 2003); types A,B,. . . From non-f.c. element can reach element with adjacent entries in τ-invariant via simple edges (move down). Observation (simply laced) x → w arc, x f.c. , w → w ′ simple = ⇒ µ(x, w) = µ(x′, w ′) for some x′

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Proof ideas

Theorem (Shi, 2003); types A,B,. . . From non-f.c. element can reach element with adjacent entries in τ-invariant via simple edges (move down). Observation (simply laced) x → w arc, x f.c. , w → w ′ simple = ⇒ µ(x, w) = µ(x′, w ′) for some x′ If x is f.c. and w is not, pull the edge to w ′ in Shi’s result; along the way guaranteed to hit edge of weight 0, 1

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Proof ideas

Theorem (Shi, 2003); types A,B,. . . From non-f.c. element can reach element with adjacent entries in τ-invariant via simple edges (move down). Observation (simply laced) x → w arc, x f.c. , w → w ′ simple = ⇒ µ(x, w) = µ(x′, w ′) for some x′ If x is f.c. and w is not, pull the edge to w ′ in Shi’s result; along the way guaranteed to hit edge of weight 0, 1 Fact (Green, 07); types A, B,. . . The statement is true for both x, w f.c. (Temperley-Lieb pictures).

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Proof ideas

Theorem (Shi, 2003); types A,B,. . . From non-f.c. element can reach element with adjacent entries in τ-invariant via simple edges (move down). Observation (simply laced) x → w arc, x f.c. , w → w ′ simple = ⇒ µ(x, w) = µ(x′, w ′) for some x′ If x is f.c. and w is not, pull the edge to w ′ in Shi’s result; along the way guaranteed to hit edge of weight 0, 1 Fact (Green, 07); types A, B,. . . The statement is true for both x, w f.c. (Temperley-Lieb pictures). Type B obstruction to pulling down: simple edges activating (1,2). Fix them using parallel transport and pattern-avoidance characterization.

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B

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Thank you!

Michael Chmutov Parallel transport in KL W -graph and Green’s 0 − 1 conjecture in type B