Z ≥ 0 -valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G). Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G).

Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅ graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Z

_≥0

-valued modules via graphs

Construct aW_∞-moduleV associated to a bipartite graphG: V=h1,2,3,4,5iC

Note that the adjacency matrixA(G) ofG is

A(G) =

Thus, in order to check that this gives aWn-module for somen we need to check that theZ≥0-valued matricesMθ_s andMθ_t

satisfy the braid-like relation ofW_n. This boils down to checking properties ofA(G).

Lusztig≤2003. The braid-like relation ofW_n is

“ ˜U_n(θ_tθ_s) = ˜U_n(θ_sθ_t)”.

Hence, by Smith’s (GP) and Lusztig: We get aZ≥0-valued module ofW_n ifG is of typeADEforn+ 1 being the Coxeter number.

Example. The Chebyshev polynomial forn= 7 is

U˜7 =X7−6X5 + 10X3−4X

The typeD₅graph has spectrum

SD5={−p The braid-like relation ofW7 is

θtθsθtθsθtθsθtθs−6θtθsθtθsθtθs+ 10θtθsθtθs−4θtθs

=θsθtθsθtθsθtθsθt−6θsθtθsθtθsθt+ 10θsθtθsθt−4θsθt.

Im Dokument Categorical representations of dihedral groups (Seite 28-41)