# Type A molecules are Kazhdan-Lusztig Michael Chmutov June 26, 2013 FPSAC ’13

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Type A molecules are Kazhdan-Lusztig Michael Chmutov June 26, 2013 FPSAC ’13
```Type A molecules are Kazhdan-Lusztig
Michael Chmutov
University of Michigan
June 26, 2013
FPSAC ’13
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
The Iwahori-Hecke Algebra and Kazhdan-Lusztig
polynomials
W = Sn , ground ring: Z[q ±1/2 ]
*
TT
= Ti+1 Ti Ti+1 , +
Ti Tj = Tj Ti ,
(Ti + 1)(Ti − q) = 0.
i
Hn =
T1 , . . . , Tn−1
i+1 Ti
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
The Iwahori-Hecke Algebra and Kazhdan-Lusztig
polynomials
W = Sn , ground ring: Z[q ±1/2 ]
*
TT
= Ti+1 Ti Ti+1 , +
Ti Tj = Tj Ti ,
(Ti + 1)(Ti − q) = 0.
i
Hn =
T1 , . . . , Tn−1
i+1 Ti
Standard basis {Tw }w ∈Sn ; Kazhdan-Lusztig basis {Cw }w ∈Sn
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
The Iwahori-Hecke Algebra and Kazhdan-Lusztig
polynomials
W = Sn , ground ring: Z[q ±1/2 ]
*
TT
= Ti+1 Ti Ti+1 , +
Ti Tj = Tj Ti ,
(Ti + 1)(Ti − q) = 0.
i
Hn =
T1 , . . . , Tn−1
i+1 Ti
Standard basis {Tw }w ∈Sn ; Kazhdan-Lusztig basis {Cw }w ∈Sn
Transition matrix entries, up to a power of q, are Kazhdan-Lusztig
polynomials Pv ,w (q)
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
The Iwahori-Hecke Algebra and Kazhdan-Lusztig
polynomials
W = Sn , ground ring: Z[q ±1/2 ]
*
TT
= Ti+1 Ti Ti+1 , +
Ti Tj = Tj Ti ,
(Ti + 1)(Ti − q) = 0.
i
Hn =
T1 , . . . , Tn−1
i+1 Ti
Standard basis {Tw }w ∈Sn ; Kazhdan-Lusztig basis {Cw }w ∈Sn
Transition matrix entries, up to a power of q, are Kazhdan-Lusztig
polynomials Pv ,w (q)
h l(w )−l(v )−1 i
)−1
2
deg (Pv ,w ) 6 l(w )−l(v
; µ(v , w ) = q
Pv ,w
2
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
(Example) Kazhdan-Lusztig W -graph
s1 s2 s1
Vertices: elements of Sn
s2 s1
s1 s2
s1
s2
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
(Example) Kazhdan-Lusztig W -graph
s1 s2 s1
Vertices: elements of Sn
s2 s1
s1 s2
Edges: weighted by µ values
s1
s2
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
(Example) Kazhdan-Lusztig W -graph
s1 s2 s1
12
Vertices: elements of Sn
s2 s1
Edges: weighted by µ values
τ -labels: left descent sets
s1
s1 s2
2
1
1
2
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
(Example) Kazhdan-Lusztig W -graph
s1 s2 s1
12
Vertices: elements of Sn
s2 s1
Edges: weighted by µ values
τ -labels: left descent sets
Directions: based on τ -label
containments
s1
s1 s2
2
1
1
2
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
(Example) Kazhdan-Lusztig W -graph
s1 s2 s1
12
Vertices: elements of Sn
s2 s1
Edges: weighted by µ values
τ -labels: left descent sets
Directions: based on τ -label
containments
s1
s1 s2
2
1
1
2
1
Fact
Easy to reconstruct KL polynomials once one has W -graph.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
(Example) Kazhdan-Lusztig W -graph
s1 s2 s1
12
Vertices: elements of Sn
s2 s1
Edges: weighted by µ values
τ -labels: left descent sets
Directions: based on τ -label
containments
s1
s1 s2
2
1
1
2
1
Fact
Easy to reconstruct KL polynomials once one has W -graph.
General Goal
Get your hands on (subgraphs of) KL graph without computing KL
polynomials.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
Outline
Kazhdan-Lusztig:
Cw
Regular repesentation
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Outline
Kazhdan-Lusztig:
Cw
KL W−graph
Michael Chmutov
Regular repesentation
Type A molecules are Kazhdan-Lusztig
Outline
Kazhdan-Lusztig:
Cw
KL W−graph
Regular repesentation
Want:
W−graph
Michael Chmutov
Repesentation
Type A molecules are Kazhdan-Lusztig
Outline
Kazhdan-Lusztig:
Cw
KL W−graph
Regular repesentation
Want:
Generalized KL polynomials
Parabolic KL polynomials
Regular KL polynomials
W−graph
Michael Chmutov
Repesentation
Type A molecules are Kazhdan-Lusztig
From graph to representation
Basis: vertices.


qu
X
Ti u = −u + q 1/2
m(u → v )v


u→v
i∈
/ τ (u)
i ∈ τ (u)
i ∈τ
/ (v )
s1 s2 s1
12
s2 s1
s1
s1 s2
2
1
1
2
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
From graph to representation
Basis: vertices.


qu
X
Ti u = −u + q 1/2
m(u → v )v


u→v
i∈
/ τ (u)
i ∈ τ (u)
i ∈τ
/ (v )
s1 s2 s1
Example
12
s2 s1
s1
s1 s2
2
w
1
1
u
2
v
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
From graph to representation
Basis: vertices.


qu
X
Ti u = −u + q 1/2
m(u → v )v


u→v
i∈
/ τ (u)
i ∈ τ (u)
i ∈τ
/ (v )
s1 s2 s1
Example
12
T1 u = qu
s2 s1
s1
s1 s2
2
w
1
1
u
2
v
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
From graph to representation
Basis: vertices.


qu
X
Ti u = −u + q 1/2
m(u → v )v


u→v
i∈
/ τ (u)
i ∈ τ (u)
i ∈τ
/ (v )
s1 s2 s1
Example
12
T1 u = qu
s2 s1
T2 u = −u + q 1/2 v + q 1/2 w
s1
s1 s2
2
w
1
1
u
2
v
1
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
s2
From graph to representation
Basis: vertices.


qu
X
Ti u = −u + q 1/2
m(u → v )v


u→v
i∈
/ τ (u)
i ∈ τ (u)
i ∈τ
/ (v )
s1 s2 s1
Example
12
T1 u = qu
s2 s1
T2 u = −u + q 1/2 v + q 1/2 w
Remark
This is how Ti ’s act on the Cw basis
with respect to KL graph.
Michael Chmutov
s1
s1 s2
2
w
1
1
u
2
v
1
Type A molecules are Kazhdan-Lusztig
s2
General W -graphs and Cells
W -graph: graph which encodes a representation via above formula
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
General W -graphs and Cells
W -graph: graph which encodes a representation via above formula
Admissible: edge-weights in Z>0 , bipartite, “edge-symmetric”
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
General W -graphs and Cells
W -graph: graph which encodes a representation via above formula
Admissible: edge-weights in Z>0 , bipartite, “edge-symmetric”
(from now on all graphs are admissible)
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
General W -graphs and Cells
W -graph: graph which encodes a representation via above formula
Admissible: edge-weights in Z>0 , bipartite, “edge-symmetric”
(from now on all graphs are admissible)
Cell: strongly connected component of W -graph
12
Michael Chmutov
2
1
1
2
Type A molecules are Kazhdan-Lusztig
General W -graphs and Cells
W -graph: graph which encodes a representation via above formula
Admissible: edge-weights in Z>0 , bipartite, “edge-symmetric”
(from now on all graphs are admissible)
Cell: strongly connected component of W -graph
12
A cell is a W -graph. (Why?)
Michael Chmutov
2
1
1
2
Type A molecules are Kazhdan-Lusztig
General W -graphs and Cells
W -graph: graph which encodes a representation via above formula
Admissible: edge-weights in Z>0 , bipartite, “edge-symmetric”
(from now on all graphs are admissible)
Cell: strongly connected component of W -graph
12
A cell is a W -graph. (Why?)
How many cells are there?
(Finitely many; Stembridge ’12)
Michael Chmutov
2
1
1
2
Type A molecules are Kazhdan-Lusztig
General W -graphs and Cells
W -graph: graph which encodes a representation via above formula
Admissible: edge-weights in Z>0 , bipartite, “edge-symmetric”
(from now on all graphs are admissible)
Cell: strongly connected component of W -graph
12
A cell is a W -graph. (Why?)
How many cells are there?
(Finitely many; Stembridge ’12)
2
1
Do all Sn cells come from the
KL graph? (Up to n = 13. . . )
1
2
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Kazhdan-Lusztig Cells
RSK correspondence:
w → (P, Q)
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Kazhdan-Lusztig Cells
RSK correspondence:
w → (P, Q)
Cell consists of w ∈ Sn with fixed Q
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Kazhdan-Lusztig Cells
RSK correspondence:
w → (P, Q)
Cell consists of w ∈ Sn with fixed Q
Cells with Qs of same shape are isomorphic
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Kazhdan-Lusztig Cells
RSK correspondence:
w → (P, Q)
Cell consists of w ∈ Sn with fixed Q
Cells with Qs of same shape are isomorphic
Simple edges, i.e. edges going in both directions, are dual Knuth
moves
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Simple edges in Kazhdan-Lusztig Cells
Examples
1
2
5
3
4
1
4
5
2
1
3
23
5
1
2
3
12
2
4
3
4
1
3
2
3
4
5
13
1
24
5
34
4
2
5
1
2
3
4
5
2
1
3
2
4
5
13
14
1
3
2
5
Michael Chmutov
4
14
1
2
3
5
4
24
Type A molecules are Kazhdan-Lusztig
1
2
4
5
3
3
Combinatorial rules
In 2008 Stembridge gave combinatorial rules for detecting when graph is
a W -graph.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Combinatorial rules
In 2008 Stembridge gave combinatorial rules for detecting when graph is
a W -graph. E.g.
1
simple edges must have weight 1,
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Combinatorial rules
In 2008 Stembridge gave combinatorial rules for detecting when graph is
a W -graph. E.g.
1
simple edges must have weight 1,
2
if (u, v ) is simple then |τ (u)∆τ (v )| 6 3.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Combinatorial rules
In 2008 Stembridge gave combinatorial rules for detecting when graph is
a W -graph. E.g.
1
simple edges must have weight 1,
2
if (u, v ) is simple then |τ (u)∆τ (v )| 6 3.
Classification of S5 cells
124
13
23
24
134
23
12
2
13
14
24
3
13
24
14
1234
123
124
134
234
1
2
3
4
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
34
Molecular components
Molecular component of a W -graph:
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Molecular components
Molecular component of a W -graph:
Fact
Each Kazhdan-Lusztig Sn cell has
only one molecular component.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main Theorem
Theorem (C., 2012)
Any molecular component of a W -graph has the same simple edges as a
Kazhdan-Lusztig one.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main Theorem
Theorem (C., 2012)
Any molecular component of a W -graph has the same simple edges as a
Kazhdan-Lusztig one.
To do
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main Theorem
Theorem (C., 2012)
Any molecular component of a W -graph has the same simple edges as a
Kazhdan-Lusztig one.
To do
Are there cells with multiple molecular components?
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main Theorem
Theorem (C., 2012)
Any molecular component of a W -graph has the same simple edges as a
Kazhdan-Lusztig one.
To do
Are there cells with multiple molecular components?
If not, are there multiple cells with a given underlying molecular
component?
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main Theorem
Theorem (C., 2012)
Any molecular component of a W -graph has the same simple edges as a
Kazhdan-Lusztig one.
To do
Are there cells with multiple molecular components?
If not, are there multiple cells with a given underlying molecular
component?
Other types?
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main ingredient: Assaf’s classification of dual equivalence
graphs
DEG: molecular component of KL cell; viewed as undirected graph.
Theorem (Assaf, 2008)
An undirected graph with labelled vertices is a DEG if and only if it
satisfies axioms (1)-(6).
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main ingredient: Assaf’s classification of dual equivalence
graphs
DEG: molecular component of KL cell; viewed as undirected graph.
Theorem (Assaf, 2008)
An undirected graph with labelled vertices is a DEG if and only if it
satisfies axioms (1)-(6).
Restricting W -graphs from Sn to
Sn−1 (or other parabolic):
Erase n − 1 from all τ -labels,
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main ingredient: Assaf’s classification of dual equivalence
graphs
DEG: molecular component of KL cell; viewed as undirected graph.
Theorem (Assaf, 2008)
An undirected graph with labelled vertices is a DEG if and only if it
satisfies axioms (1)-(6).
Restricting W -graphs from Sn to
Sn−1 (or other parabolic):
Erase n − 1 from all τ -labels,
on τ -label containment
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Main ingredient: Assaf’s classification of dual equivalence
graphs
DEG: molecular component of KL cell; viewed as undirected graph.
Theorem (Assaf, 2008)
An undirected graph with labelled vertices is a DEG if and only if it
satisfies axioms (1)-(6).
Axiom 6; in molecular language
Restricting W -graphs from Sn to
Sn−1 (or other parabolic):
Erase n − 1 from all τ -labels,
on τ -label containment
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Prove that if axiom 6 does not
hold then restrictions are
inconsistent.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Prove that if axiom 6 does not
hold then restrictions are
inconsistent.
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Prove that if axiom 6 does not
hold then restrictions are
inconsistent.
Michael Chmutov
simple
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Prove that if axiom 6 does not
hold then restrictions are
inconsistent.
Michael Chmutov
1
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Prove that if axiom 6 does not
hold then restrictions are
inconsistent.
Michael Chmutov
1
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Prove that if axiom 6 does not
hold then restrictions are
inconsistent.
Michael Chmutov
1
Type A molecules are Kazhdan-Lusztig
Proof idea
Strategy:
Classify all S5 cells and ways
they can bind. Similarly for
S3 × S2 .
1
This yields rules, e.g.:
23
12
13
24
34
24
134
14
124
13
23
Prove that if axiom 6 does not
hold then restrictions are
inconsistent.
Michael Chmutov
1
Type A molecules are Kazhdan-Lusztig
Thank you!
Michael Chmutov
Type A molecules are Kazhdan-Lusztig
```
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