COMBINATORIAL INTERPRETATION OF HECKE ALGEBRA TRACES Sam Clearman, - - PowerPoint PPT Presentation

COMBINATORIAL INTERPRETATION OF HECKE ALGEBRA TRACES

Sam Clearman, Matthew Hyatt, Brittany Shelton, and Mark Skandera Lehigh University

Outline (1) The Hecke algebra and its traces (2) The chromatic symmetric and quasisymmetric functions (3) Zig-zag networks (4) Formula for evaluating induced trivial characters (5) Conjectured formula for evaluating power sum traces

The Hecke algebra Hn(q) Generators over C[q

1 2, q

¯1

2]: Ts1, . . . , Tsn−1.

Relations: T 2

si = (q − 1)Tsi + qTe

for i = 1, . . . , n − 1, TsiTsjTsi = TsjTsiTsj for |i − j| = 1, TsiTsj = TsjTsi for |i − j| ≥ 2. Natural basis: {Tw | w ∈ Sn}, Tw = Tsi1 · · · Tsiℓ, (w = si1 · · · siℓ reduced). Kazhdan-Lusztig basis: {C′

w | w ∈ Sn},

C′

w = q

ℓ(w) 2

• v≤w

Pv,w(q)Tv. Call θq : Hn(q) → C[q

1 2, q

¯1

2] a trace if θq(gh) = θq(hg) ∀g, h.

Examples of Hecke algebra traces Irreducible characters: {χλ

q | λ ⊢ n}.

Induced trivial, sign characters: {ηλ

q | λ ⊢ n}, {ǫλ q | λ ⊢ n}.

Monomial, power sum traces: {φλ

q | λ ⊢ n}, {ψλ q | λ ⊢ n}.

ηλ

q =

• µ

Kµ,λχµ

q,

ǫλ

q =

• µ

Kµ

⊤

,λχµ q,

φλ

q =

• µ

K−1

λ,µχµ q,

ψλ

q =

• µ

Lλ,µφµ

q.

hλ =

• µ

Kµ,λsµ, eλ =

• µ

Kµ

⊤

,λsµ,

mλ =

• µ

K−1

λ,µsµ,

pλ =

• µ

Lλ,µmµ.

Formulas for trace evaluations θq θq(Tw) in N[q]? interpretation of θq(Tw) as

• k

(−1)|Sk||Rk|qk ? θq(q

ℓ(w) 2 C′

w)

in N[q]? interpretation of θq(q

ℓ(w) 2 C′

w)

as

• k

|Rk|qk for w avoiding 3412, 4231? ηλ

q

no

• pen

H ’92 CHSS ’13 ǫλ

q

no

• pen

H ’92 CHSS ’12 χλ

q

no

• pen

H ’92 CHSS ’13 ψλ

q

no

• pen

(conj. H ’92) conj. CHSS ’13 φλ

q

no

• pen
• conj. H ’92
• pen

Connection to chromatic symmetric functions Stanley (’95) associated to each poset P a chromatic symmetric func- tion XP. P an n-element unit interval order = ⇒ ∃w ∈ Sn avoiding 312 s.t. XP =

• λ⊢n

ǫλ(C′

w(1))mλ =

• λ⊢n

ηλ(C′

w(1))fλ =

• λ⊢n

χλ

⊤

(C′

w(1))sλ

=

• λ⊢n

ψλ(C′

w(1))

(−1)n−ℓ(λ) pλ zλ =

• λ⊢n

φλ(C′

w(1))eλ.

Conj: (SS ’93) For P a unit interval order, XP ∈ spanN{eλ | λ ⊢ n}. (c.f. H ’92.)

Connection to chromatic quasisymmetric functions Shareshian, Wachs (’12) introduced a quasisymmetric q-analog XP,q

• f XP.

Thm: (CHSS ’13) For P an appropriately labeled unit interval order, ∃w ∈ Sn avoiding 312 s.t. XP,q =

• λ⊢n

ǫλ

q(q

ℓ(w) 2 C′

w)mλ =

• λ⊢n

ηλ

q (q

ℓ(w) 2 C′

w)fλ =

• λ⊢n

χλ

⊤

q (q

ℓ(w) 2 C′

w)sλ

=

• λ⊢n

ψλ

q (q

ℓ(w) 2 C′

w)

(−1)n−ℓ(λ) pλ zλ =

• λ⊢n

φλ

q(q

ℓ(w) 2 C′

w)eλ.

Conj: (SW ’12) For P an appropriately labeled unit interval order, XP,q ∈ spanN[q]{eλ | λ ⊢ n}. (c.f. H ’92.)

Zig-zag networks Thm: (S ’08) For w∈Sn avoiding 3412, 4231, the element C′

w ∈Hn(q) can be encoded by a zig-zag network of order n.

Ex: C′

258431976 is encoded by the zig-zag network (c.f. BW ’01)

→ →

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Call vertices on left sources, vertices on right sinks. Special case: w ∈ Sn avoids 312.

Path families and F-tableaux Call a sequence of paths from sources 1, . . . , n to sinks w1, . . . , wn a path family of type w. Example: F =

1 2 3 4 1 2 3 4

. path families:

• 1

2 3 4 1 2 3 4

,

1 2 3 4 1 2 3 4

,

1 2 3 4 1 2 3 4

, . . .

• types :

1234 1234

• ,

1234 2134

• ,

1234 3421

• , . . .
• .

There is at most one path from source i to sink j. Call it (i, j). Define an F-tableau of shape λ to be a placement of a path family ((1, w1), . . . , (n, wn)) into a Young diagram of shape λ.

Row-closed and left row-strict tableaux Call an F-tableau row-closed if in each row, the sets of source and sink indices are equal. Call it left row-strict if source indices increase to the right. Ex: Two such tableaux of shape 31 for F =

1 2 3 4 1 2 3 4

are

3,3 1,2 4,1 2,4

,

1,3 3,2 2,1 4,4

. Let Uk denote row k of tableau U. Let ◦ denote concatenation. For each tableau U above, U1 ◦ U2 is equal to

1,2 4,1 2,4 3,3 , 1,3 3,2 2,1 4,4 .

Right inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a right inversion in an F-tableau if wi > wj and (i, wi) appears earlier than (j, wj). Let rinv(U) denote the number of right inversions in U. The previous tableaux satisfy rinv(U1 ◦ U2) = 3, 2, respectively: F =

1 2 3 4 1 2 3 4

,

3,3 1,2 2,4 4,1

,

1,3 3,2 2,1 4,4 .

Right inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a right inversion in an F-tableau if wi > wj and (i, wi) appears earlier than (j, wj). Let rinv(U) denote the number of right inversions in U. The previous tableaux satisfy rinv(U1 ◦ U2) = 3, 2, respectively: F =

1 2 3 4 1 2 3 4

,

3,3 1,2 2,4 4,1

,

1,3 3,2 2,1 4,4 .

Right inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a right inversion in an F-tableau if wi > wj and (i, wi) appears earlier than (j, wj). Let rinv(U) denote the number of right inversions in U. The previous tableaux satisfy rinv(U1 ◦ U2) = 3, 2, respectively: F =

1 2 3 4 1 2 3 4

,

3,3 1,2 2,4 4,1

,

1,3 3,2 2,1 4,4 .

Right inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a right inversion in an F-tableau if wi > wj and (i, wi) appears earlier than (j, wj). Let rinv(U) denote the number of right inversions in U. The previous tableaux satisfy rinv(U1 ◦ U2) = 3, 2, respectively: F =

1 2 3 4 1 2 3 4

,

1,2 4,1 2,4 3,3 , 4,4 1,3 2,1 3,2

.

Right inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a right inversion in an F-tableau if wi > wj and (i, wi) appears earlier than (j, wj). Let rinv(U) denote the number of right inversions in U. The previous tableaux satisfy rinv(U1 ◦ U2) = 3, 2, respectively: F =

1 2 3 4 1 2 3 4

,

1,2 4,1 2,4 3,3 , 4,4 1,3 2,1 3,2

.

Induced trivial characters Thm: (CHSS ’13) For w avoiding 3412, 4231, we have ηλ

q (q

ℓ(w) 2 C′

w) =

• U

qrinv(U1◦···◦Ur), where the sum is over all row-closed, left row-strict F-tableaux of shape λ = (λ1, . . . , λr), and F corresponds to w. Ex: For previous network F, we have w = 3421 and η4

q(q

ℓ(w) 2 C′

w) = 1 + 3q + 5q2 + 5q3 + 3q4 + q5,

η31

q (q

ℓ(w) 2 C′

w) = 1 + 3q + 6q2 + 6q3 + 3q4 + q5,

η22

q (q

ℓ(w) 2 C′

w) = 1 + 3q + 6q2 + 6q3 + 3q4 + q5,

η211

q

(q

ℓ(w) 2 C′

w) = 1 + 3q + 7q2 + 7q3 + 3q4 + q5,

η1111

q

(q

ℓ(w) 2 C′

w) = 1 + 3q + 8q2 + 8q3 + 3q4 + q5.

Cylindrical tableaux Call an F-tableau cylindrical if each row has the form (i1, i2), (i2, i3), . . . , (ik, i1). Example: Two such tableaux of shape 31 for F =

1 2 3 4 1 2 3 4

are

3,3 1,2 4,1 2,4

,

4,4 2,3 1,2 3,1

. For each tableau U above, U2 ◦ U1 is equal to

4,1 3,3 1,2 2,4

,

1,2 4,4 2,3 3,1

.

Inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a (left) inversion in an F-tableau if i > j and (i, wi) appears earlier than (j, wj). Let inv(U) denote the number of inversions in U. The previous tableaux satisfy inv(U2 ◦ U1) = 2, 4, respectively: F =

1 2 3 4 1 2 3 4

,

4,1 3,3 1,2 2,4

,

1,2 4,4 2,3 3,1

.

Inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a (left) inversion in an F-tableau if i > j and (i, wi) appears earlier than (j, wj). Let inv(U) denote the number of inversions in U. The previous tableaux satisfy inv(U2 ◦ U1) = 2, 4, respectively: F =

1 2 3 4 1 2 3 4

,

4,1 2,4 1,2 3,3

,

1,2 4,4 2,3 3,1

.

Inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a (left) inversion in an F-tableau if i > j and (i, wi) appears earlier than (j, wj). Let inv(U) denote the number of inversions in U. The previous tableaux satisfy inv(U2 ◦ U1) = 2, 4, respectively: F =

1 2 3 4 1 2 3 4

,

4,1 3,3 1,2 2,4

,

1,2 4,4 2,3 3,1

.

Inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a (left) inversion in an F-tableau if i > j and (i, wi) appears earlier than (j, wj). Let inv(U) denote the number of inversions in U. The previous tableaux satisfy inv(U2 ◦ U1) = 2, 4, respectively: F =

1 2 3 4 1 2 3 4

,

4,1 3,3 1,2 2,4

,

1,2 4,4 2,3 3,1

.

Inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a (left) inversion in an F-tableau if i > j and (i, wi) appears earlier than (j, wj). Let inv(U) denote the number of inversions in U. The previous tableaux satisfy inv(U2 ◦ U1) = 2, 4, respectively: F =

1 2 3 4 1 2 3 4

,

4,1 3,3 1,2 2,4

,

1,2 4,4 2,3 3,1

NO

.

Inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a (left) inversion in an F-tableau if i > j and (i, wi) appears earlier than (j, wj). Let inv(U) denote the number of inversions in U. The previous tableaux satisfy inv(U2 ◦ U1) = 2, 4, respectively: F =

1 2 3 4 1 2 3 4

,

4,1 3,3 1,2 2,4

,

1,2 4,4 2,3 3,1

.

Inversions in F-tableaux Call intersecting paths (i, wi) and (j, wj) a (left) inversion in an F-tableau if i > j and (i, wi) appears earlier than (j, wj). Let inv(U) denote the number of inversions in U. The previous tableaux satisfy inv(U2 ◦ U1) = 2, 4, respectively: F =

1 2 3 4 1 2 3 4

,

4,1 3,3 1,2 2,4

,

1,2 4,4 2,3 3,1

.

Power sum traces Conj: (CHSS ’13) For w avoiding 3412, 4231, we have ψλ

q (q

ℓ(w) 2 C′

w) =

• U

qinv(Ur◦···◦U1), where F corresponds to w, and the sum is over all cylindrical F- tableaux of shape λ = (λ1, . . . , λr). Ex: For previous network F, we have w = 3421 and ψ4

q(q

ℓ(w) 2 C′

w) = 1 + 3q + 4q2 + 4q3 + 3q4 + q5,

ψ31

q (q

ℓ(w) 2 C′

w) = 1 + 3q + 5q2 + 5q3 + 3q4 + q5,

ψ22

q (q

ℓ(w) 2 C′

w) = 1 + 3q + 4q2 + 4q3 + 3q4 + q5,

ψ211

q

(q

ℓ(w) 2 C′

w) = 1 + 3q + 6q2 + 6q3 + 3q4 + q5,

ψ1111

q

(q

ℓ(w) 2 C′

w) = 1 + 3q + 8q2 + 8q3 + 3q4 + q5.

Open questions Shareshian-Wachs conjectured interpretations of coefficients of pλ in XP,q. By main theorem, these should be equal to the conjectured evaluations of ψλ

q .

Question: Are the conjectures equivalent? Question: Is there a similar combinatorial interpretation of φλ

q(q

ℓ(w) 2 C′

w)

when w avoids 3412, 4231? Conj: (SS ’93) We have φλ(C′

w(1)) ≥ 0 when w avoids 3412, 4231.

Conj: (H ’92) We have φλ(q

ℓ(w) 2 C′

w) ∈ N[q] for all w.

Conj: (SW ’12) We have φλ(q

ℓ(w) 2 C′

w) ∈ N[q] when w avoids 3412,

4231.