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 H n ( q ) 1 ¯ 1 Generators over C [ q 2 ]: T s 1 , . . . , T s n − 1 . 2 , q Relations: T 2 s i = ( q − 1) T s i + qT e for i = 1 , . . . , n − 1 , T s i T s j T s i = T s j T s i T s j for | i − j | = 1 , T s i T s j = T s j T s i for | i − j | ≥ 2 . Natural basis: { T w | w ∈ S n } , T w = T s i 1 · · · T s iℓ , ( w = s i 1 · · · s i ℓ reduced) . Kazhdan-Lusztig basis: { C ′ w | w ∈ S n } , ℓ ( w ) C ′ � w = q P v,w ( q ) T v . 2 v ≤ w 1 ¯ 1 Call θ q : H n ( q ) → C [ q 2 ] a trace if θ q ( gh ) = θ q ( hg ) ∀ g, h . 2 , q

Examples of Hecke algebra traces Irreducible characters: { χ λ q | λ ⊢ n } . Induced trivial, sign characters: { η λ q | λ ⊢ n } , { ǫ λ q | λ ⊢ n } . Monomial, power sum traces: { φ λ q | λ ⊢ n } , { ψ λ q | λ ⊢ n } . K µ,λ χ µ ,λ χ µ λ,µ χ µ K − 1 η λ ǫ λ φ λ � � � q = q = q = q , K µ q , q , ⊤ µ µ µ L λ,µ φ µ ψ λ � q = q . µ K − 1 � � � h λ = e λ = m λ = K µ,λ s µ , K µ ,λ s µ , λ,µ s µ , ⊤ µ µ µ � p λ = L λ,µ m µ . µ

Formulas for trace evaluations interpretation of ℓ ( w ) interpretation of 2 C ′ θ q ( q w ) ℓ ( w ) θ q ( T w ) θ q ( T w ) 2 C ′ as θ q ( q w ) in as θ q � | R k | q k in ( − 1) | S k | | R k | q k ? � N [ q ]? N [ q ]? k for w avoiding k 3412, 4231? η λ no open H ’92 CHSS ’13 q ǫ λ no H ’92 CHSS ’12 open q χ λ no open H ’92 CHSS ’13 q ψ λ no open (conj. H ’92) conj. CHSS ’13 q φ λ no open open conj. H ’92 q

Connection to chromatic symmetric functions Stanley (’95) associated to each poset P a chromatic symmetric func- tion X P . P an n -element unit interval order = ⇒ ∃ w ∈ S n avoiding 312 s.t. ⊤ ǫ λ ( C ′ η λ ( C ′ ( C ′ � � � χ λ X P = w (1)) m λ = w (1)) f λ = w (1)) s λ λ ⊢ n λ ⊢ n λ ⊢ n ψ λ ( C ′ w (1)) p λ φ λ ( C ′ � � = = w (1)) e λ . ( − 1) n − ℓ ( λ ) z λ λ ⊢ n λ ⊢ n Conj: (SS ’93) For P a unit interval order, X P ∈ span N { e λ | λ ⊢ n } . (c.f. H ’92.)

Connection to chromatic quasisymmetric functions Shareshian, Wachs (’12) introduced a quasisymmetric q -analog X P,q of X P . Thm: (CHSS ’13) For P an appropriately labeled unit interval order, ∃ w ∈ S n avoiding 312 s.t. ℓ ( w ) ℓ ( w ) ℓ ( w ) ⊤ 2 C ′ 2 C ′ 2 C ′ � ǫ λ � η λ � χ λ X P,q = q ( q w ) m λ = q ( q w ) f λ = q ( q w ) s λ λ ⊢ n λ ⊢ n λ ⊢ n ℓ ( w ) 2 C ′ ψ λ q ( q w ) p λ ℓ ( w ) 2 C ′ � � φ λ = = q ( q w ) e λ . ( − 1) n − ℓ ( λ ) z λ λ ⊢ n λ ⊢ n Conj: (SW ’12) For P an appropriately labeled unit interval order, X P,q ∈ span N [ q ] { e λ | λ ⊢ n } . (c.f. H ’92.)

Zig-zag networks Thm: (S ’08) For w ∈ S n avoiding 3412 , 4231, the element C ′ w ∈ H n ( 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) 9 9 8 8 7 7 6 6 5 5 → → 4 4 3 3 2 2 1 1 Call vertices on left sources , vertices on right sinks . Special case: w ∈ S n avoids 312.

Path families and F -tableaux Call a sequence of paths from sources 1 , . . . , n to sinks w 1 , . . . , w n a path family of type w . 4 4 3 3 Example: F = . 2 2 1 1 � 4 4 4 4 4 4 � 3 3 3 3 3 3 path families: , , , . . . 2 2 2 2 2 2 1 1 1 1 1 1 �� 1234 � � � 1234 � � 1234 � types : , , , . . . . 1234 2134 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 , w 1 ) , . . . , ( n, w n )) 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. 4 4 3 3 Ex: Two such tableaux of shape 31 for F = are 2 2 1 1 3,3 4,4 1,2 2,4 4,1 1,3 2,1 3,2 , . Let U k denote row k of tableau U . Let ◦ denote concatenation. For each tableau U above, U 1 ◦ U 2 is equal to 3,3 , 4,4 . 1,2 2,4 4,1 1,3 2,1 3,2

Right inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a right inversion in an F -tableau if w i > w j and ( i, w i ) appears earlier than ( j, w j ). Let rinv ( U ) denote the number of right inversions in U . The previous tableaux satisfy rinv ( U 1 ◦ U 2 ) = 3 , 2, respectively: 4 4 3 3 F = , , 4,4 . 1,2 2,4 4,1 3,3 1,3 2,1 3,2 2 2 1 1

Right inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a right inversion in an F -tableau if w i > w j and ( i, w i ) appears earlier than ( j, w j ). Let rinv ( U ) denote the number of right inversions in U . The previous tableaux satisfy rinv ( U 1 ◦ U 2 ) = 3 , 2, respectively: 4 4 3 3 F = , , 4,4 . 1,2 2,4 4,1 3,3 1,3 2,1 3,2 2 2 1 1

Right inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a right inversion in an F -tableau if w i > w j and ( i, w i ) appears earlier than ( j, w j ). Let rinv ( U ) denote the number of right inversions in U . The previous tableaux satisfy rinv ( U 1 ◦ U 2 ) = 3 , 2, respectively: 4 4 3 3 F = , , 4,4 . 1,2 2,4 4,1 3,3 1,3 2,1 3,2 2 2 1 1

Right inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a right inversion in an F -tableau if w i > w j and ( i, w i ) appears earlier than ( j, w j ). Let rinv ( U ) denote the number of right inversions in U . The previous tableaux satisfy rinv ( U 1 ◦ U 2 ) = 3 , 2, respectively: 4 4 3 3 F = , 3,3 , . 1,2 2,4 4,1 1,3 2,1 3,2 4,4 2 2 1 1

Right inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a right inversion in an F -tableau if w i > w j and ( i, w i ) appears earlier than ( j, w j ). Let rinv ( U ) denote the number of right inversions in U . The previous tableaux satisfy rinv ( U 1 ◦ U 2 ) = 3 , 2, respectively: 4 4 3 3 F = , 3,3 , . 1,2 2,4 4,1 1,3 2,1 3,2 4,4 2 2 1 1

Induced trivial characters Thm: (CHSS ’13) For w avoiding 3412, 4231, we have ℓ ( w ) 2 C ′ q rinv ( U 1 ◦···◦ U r ) , η λ � q ( q w ) = U where the sum is over all row-closed, left row-strict F -tableaux of shape λ = ( λ 1 , . . . , λ r ), and F corresponds to w . Ex: For previous network F , we have w = 3421 and ℓ ( w ) w ) = 1 + 3 q + 5 q 2 + 5 q 3 + 3 q 4 + q 5 , η 4 2 C ′ q ( q ℓ ( w ) w ) = 1 + 3 q + 6 q 2 + 6 q 3 + 3 q 4 + q 5 , η 31 2 C ′ q ( q ℓ ( w ) w ) = 1 + 3 q + 6 q 2 + 6 q 3 + 3 q 4 + q 5 , η 22 2 C ′ q ( q ℓ ( w ) w ) = 1 + 3 q + 7 q 2 + 7 q 3 + 3 q 4 + q 5 , η 211 2 C ′ ( q q ℓ ( w ) w ) = 1 + 3 q + 8 q 2 + 8 q 3 + 3 q 4 + q 5 . η 1111 2 C ′ ( q q

Cylindrical tableaux Call an F -tableau cylindrical if each row has the form ( i 1 , i 2 ) , ( i 2 , i 3 ) , . . . , ( i k , i 1 ) . 4 4 3 3 Example: Two such tableaux of shape 31 for F = are 2 2 1 1 3,3 4,4 1,2 2,4 4,1 2,3 3,1 1,2 , . For each tableau U above, U 2 ◦ U 1 is equal to 3,3 1,2 2,4 4,1 4,4 2,3 3,1 1,2 , .

Inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a (left) inversion in an F -tableau if i > j and ( i, w i ) appears earlier than ( j, w j ). Let inv ( U ) denote the number of inversions in U . The previous tableaux satisfy inv ( U 2 ◦ U 1 ) = 2 , 4, respectively: 4 4 3 3 F = , , . 3,3 1,2 2,4 4,1 4,4 2,3 3,1 1,2 2 2 1 1

Inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a (left) inversion in an F -tableau if i > j and ( i, w i ) appears earlier than ( j, w j ). Let inv ( U ) denote the number of inversions in U . The previous tableaux satisfy inv ( U 2 ◦ U 1 ) = 2 , 4, respectively: 4 4 3 3 F = , , . 3,3 1,2 2,4 4,1 4,4 2,3 3,1 1,2 2 2 1 1

Inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a (left) inversion in an F -tableau if i > j and ( i, w i ) appears earlier than ( j, w j ). Let inv ( U ) denote the number of inversions in U . The previous tableaux satisfy inv ( U 2 ◦ U 1 ) = 2 , 4, respectively: 4 4 3 3 F = , , . 3,3 1,2 2,4 4,1 4,4 2,3 3,1 1,2 2 2 1 1

Inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a (left) inversion in an F -tableau if i > j and ( i, w i ) appears earlier than ( j, w j ). Let inv ( U ) denote the number of inversions in U . The previous tableaux satisfy inv ( U 2 ◦ U 1 ) = 2 , 4, respectively: 4 4 3 3 F = , , . 3,3 1,2 2,4 4,1 4,4 2,3 3,1 1,2 2 2 1 1

Inversions in F -tableaux Call intersecting paths ( i, w i ) and ( j, w j ) a (left) inversion in an F -tableau if i > j and ( i, w i ) appears earlier than ( j, w j ). Let inv ( U ) denote the number of inversions in U . The previous tableaux satisfy inv ( U 2 ◦ U 1 ) = 2 , 4, respectively: 4 4 3 3 F = , , . 3,3 1,2 2,4 4,1 4,4 2,3 3,1 1,2 2 2 1 1 NO