Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Philosophy.
Imagine a graph whose vertices are thex’s or them’s. v1→v2ifv1appears inzv2.
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”. Group algebras with the group element basis have only one cell,G itself. TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979). Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux. Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs
t st tst stst tstst w0
We will see the transitiveN0-modules in a second.
Left cells Right cells Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Philosophy.
Imagine a graph whose vertices are thex’s or them’s.
v1→v2ifv1appears inzv2.
x1
x2
x3
x4 m1
m2
m3
m4
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Example.
Group algebras with the group element basis have only one cell,G itself. TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979). Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux. Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs
t st tst stst tstst w0
We will see the transitiveN0-modules in a second.
Left cells Right cells Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Philosophy.
Imagine a graph whose vertices are thex’s or them’s.
v1→v2ifv1appears inzv2.
x1
x2
x3
x4 m1
m2
m3
m4
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Group algebras with the group element basis have only one cell,G itself. TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979). Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux. Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs
t st tst stst tstst w0
We will see the transitiveN0-modules in a second.
Left cells Right cells Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Philosophy.
Imagine a graph whose vertices are thex’s or them’s.
v1→v2ifv1appears inzv2.
x1
x2
x3
x4 m1
m2
m3
m4
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Example.
Group algebras with the group element basis have only one cell,G itself. TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979). Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux. Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs
t st tst stst tstst w0
We will see the transitiveN0-modules in a second.
Left cells Right cells Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Imagine a graph whose vertices are thex’s or them’s. v1→v2ifv1appears inzv2.
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Example.
Group algebras with the group element basis have only one cell,G itself.
TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979). Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux. Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs
t st tst stst tstst w0
We will see the transitiveN0-modules in a second.
Left cells Right cells Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Philosophy.
Imagine a graph whose vertices are thex’s or them’s. v1→v2ifv1appears inzv2.
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Example.
Group algebras with the group element basis have only one cell,G itself.
TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979).
Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux.
Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs
t st tst stst tstst w0
We will see the transitiveN0-modules in a second.
Left cells Right cells Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Imagine a graph whose vertices are thex’s or them’s. v1→v2ifv1appears inzv2.
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Example.
Group algebras with the group element basis have only one cell,G itself.
TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979).
Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux.
Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs w
Left cells
Right cells Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Philosophy.
Imagine a graph whose vertices are thex’s or them’s. v1→v2ifv1appears inzv2.
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Example.
Group algebras with the group element basis have only one cell,G itself.
TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979).
Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux.
Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs w0
Left cells
Right cells
Two-sided cells
Cells of N
0-algebras and N
0-modules
Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈BP. x∼Lyif x≤Lyandy≤Lx.
∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.
AnN0-module Mis transitive if all basis elements belong to the same∼L equivalence class. An apex ofMis a maximal two-sided cell not killing it.
Fact. Each transitiveN0-module has a unique apex.
Example. TransitiveN0-modules arise as decategorifications of simple 2-modules.
Imagine a graph whose vertices are thex’s or them’s. v1→v2ifv1appears inzv2.
cells = connected components transitive = one connected component
“The basic building blocks ofN0-representation theory”.
Example.
Group algebras with the group element basis have only one cell,G itself.
TransitiveN0-modules areC[G/H] forH being a subgroup. The apex isG.
Example (Kazhdan–Lusztig∼1979).
Hecke algebras for the symmetric group with KL basis
have cells coming from the Robinson–Schensted correspondence.
The transitiveN0-modules are the simples
with apex given by elements for the same shape of Young tableaux.
Example (Lusztig ≤2003).
Hecke algebras for the dihedral group with KL basis have the following cells:
1
s ts sts tsts ststs w
Left cells Right cells
Two-sided cells