Weighted walks around dissected polygons Conway-Coxeter friezes and - PowerPoint PPT Presentation

Weighted walks around dissected polygons – Conway-Coxeter friezes and beyond Christine Bessenrodt MIT, June 27, 2014

Weighted walks around dissected polygons – Conway-Coxeter friezes and beyond Christine Bessenrodt 1 1 1 0 2 1 0 0 4

Weighted walks around dissected polygons – Conway-Coxeter friezes and beyond Christine Bessenrodt 1 1 1 0 2 1 0 0 4 7

Weighted walks around dissected polygons – Conway-Coxeter friezes and beyond Christine Bessenrodt 1 2 1 2 2 1 0 4 2 1 0 0 7 2

Weighted walks around dissected polygons – Conway-Coxeter friezes and beyond Christine Bessenrodt 1 2 1 2 2 1 0 4 2 1 0 0 7 2 70

Weighted walks around dissected polygons – Conway-Coxeter friezes and beyond Christine Bessenrodt 1 2 1 2 2 1 0 4 2 1 0 0 7 2 Stanley@70 Happy Birthday, Richard!

Arithmetical friezes Conway, Coxeter (1973) 0 0 0 0 0 0 0 0 . . . . . . . . . 1 1 1 1 1 1 1 1 1 . . . · · · · · · · · . . . . . . · · · · · · · · · . . . . . . · · · · · · · . . . b . . . · · · · · · · . . . a d . . . · · · · · · · . . . c . . . . . . · · · · · · · · · . . . · · · · · · · · . . . . . . . . . 1 1 1 1 1 1 1 1 1 . . . 0 0 0 0 0 0 0 0 . . . . . . a,b,c,d ∈ N , ad − bc = 1

Conway-Coxeter friezes A frieze pattern of height 4: 1 1 1 1 1 1 1 1 1 . . . . . . . . . 1 3 1 2 2 1 3 1 2 . . . 2 2 1 3 1 2 2 1 3 . . . . . . . . . 1 1 1 1 1 1 1 1 1 . . .

Conway-Coxeter friezes A frieze pattern of height 4: 1 1 1 1 1 1 1 1 1 . . . . . . . . . 1 3 1 2 2 1 3 1 2 . . . 2 2 1 3 1 2 2 1 3 . . . . . . . . . 1 1 1 1 1 1 1 1 1 . . . · · · · · ·

Classification of friezes via triangulated polygons . . . 1 1 1 1 1 1 1 1 1 . . . 1 3 1 2 2 1 3 1 2 . . . . . . 2 2 1 3 1 2 2 1 3 . . . . . . 1 1 1 1 1 1 1 1 1 . . . . . .

Classification of friezes via triangulated polygons 1 1 1 1 1 1 1 1 1 . . . . . . 1 3 1 2 2 1 3 1 2 . . . . . . . . . 2 2 1 3 1 2 2 1 3 . . . 1 1 1 1 1 1 1 1 1 . . . . . . �

Classification of friezes via triangulated polygons 1 1 1 1 1 1 1 1 1 . . . . . . . . . 1 3 1 2 2 1 3 1 2 . . . 2 2 1 3 1 2 2 1 3 . . . . . . . . . 1 1 1 1 1 1 1 1 1 . . . � 1 3 2 1 2 Count number of triangles at each vertex!

Arcs Broline, Crowe, Isaacs (1974) 1 α 2 5 β γ 3 4 An arc from vertex i to vertex j is a sequence of different triangles ( t i + 1 , t i + 2 , . . . , t j − 1 ) such that t k is incident to vertex k , for all k . arcs 1 2 3 4 5 from 1 to - ∅ ( α ) , ( β ) , ( γ ) ( α, γ ) , ( β, γ ) ( α, γ, β )

Arcs Broline, Crowe, Isaacs (1974) 1 α 2 5 β γ 3 4 An arc from vertex i to vertex j is a sequence of different triangles such that t k is incident to vertex k , for all k . ( t i + 1 , t i + 2 , . . . , t j − 1 ) arcs 1 2 3 4 5 from 1 to - ∅ ( α ) , ( β ) , ( γ ) ( α, γ ) , ( β, γ ) ( α, γ, β ) count! 0 1 3 2 1

Arc enumeration 1 α 2 5 β  0 1 3 2 1  γ 1 0 1 1 1   3 4   W = 3 1 0 1 2     2 1 1 0 1   1 1 2 1 0

Arc enumeration – and back to the frieze 1 α 2 5 β   0 1 3 2 1 γ 1 0 1 1 1   3 4   W = 3 1 0 1 2     2 1 1 0 1   1 1 2 1 0 1 1 1 1 1 1 1 1 1 . . . . . . . . . 1 3 1 2 2 1 3 1 2 . . . 2 2 1 3 1 2 2 1 3 . . . . . . . . . 1 1 1 1 1 1 1 1 1 . . .

Arc enumeration – and back to the frieze 1 α 2 5 β  0 1 3 2 1  γ 1 0 1 1 1   3 4   W = 3 1 0 1 2     2 1 1 0 1   1 1 2 1 0 1 1 1 1 1 1 1 1 1 . . . . . . . . . 1 3 1 2 2 1 3 1 2 . . . 2 2 1 3 1 2 2 1 3 . . . . . . 1 1 1 1 1 1 1 1 1 . . . . . .

The arc enumeration matrix 1   0 1 3 2 1 α 1 0 1 1 1 2 5     W = 3 1 0 1 2 det W = 8     2 1 1 0 1 β   γ 1 1 2 1 0 3 4

The arc enumeration matrix – the frieze table 1   0 1 3 2 1 α 1 0 1 1 1 2 5     W = 3 1 0 1 2 det W = 8     2 1 1 0 1 β   γ 1 1 2 1 0 3 4 Theorem (Broline, Crowe, Isaacs 1974) Let W be the arc enumeration matrix to a triangulated n -gon. (i) W is a symmetric matrix, with its upper/lower part equal to the fundamental domain of the frieze to the triangulation. (ii) det W = −(− 2 ) n − 2 .

Renewed interest: recent generalizations and refinements Remarks 1 Frieze patterns in the context of cluster algebras of type A! Caldero, Chapoton; Propp; Assem, Dupont, Reutenauer, Schiffler, Smith; Baur, Marsh; Morier-Genoud, Ovsienko, Tabachnikov; Holm, Jørgensen, ...

Renewed interest: recent generalizations and refinements Remarks 1 Frieze patterns in the context of cluster algebras of type A! Caldero, Chapoton; Propp; Assem, Dupont, Reutenauer, Schiffler, Smith; Baur, Marsh; Morier-Genoud, Ovsienko, Tabachnikov; Holm, Jørgensen, ... 2 Generalization to d -angulations and a refinement giving the Smith normal form of the corresponding “frieze table”. In this context, a generalized frieze pattern is associated to the d -angulation where the local 2 × 2 determinants are 0 or 1. (Joint work with Thorsten Holm and Peter Jørgensen, JCTA 2014.)

Weighted arcs 1 α 2 5 β γ 3 4 arcs 1 2 3 4 5 from 1 to - ∅ ( α ) , ( β ) , ( γ ) ( α, γ ) , ( β, γ ) ( α, γ, β )

Weighted arcs 1 α 2 5 β γ 3 4 arcs 1 2 3 4 5 from 1 to - ∅ ( α ) , ( β ) , ( γ ) ( α, γ ) , ( β, γ ) ( α, γ, β ) weights! 0 1 a+b+c ac+bc abc  0 1 a + b + c ac + bc  abc abc 0 1 c bc     0 1 W = ab + ac + bc abc b + c     a + b ab abc 0 1   1 a ab + ac abc 0

Weighted arcs 1 α 2 5 β γ 3 4 arcs 1 2 3 4 5 from 1 to - ∅ ( α ) , ( β ) , ( γ ) ( α, γ ) , ( β, γ ) ( α, γ, β ) weights! 0 1 a+b+c ac+bc abc  0 1 a + b + c ac + bc  abc abc 0 1 c bc     0 1 W = ab + ac + bc abc b + c     a + b ab abc 0 1   1 a ab + ac abc 0 det W = a 5 b 5 c 5 + a 4 b 2 c 4 + a 4 b 4 c 2 + a 2 b 4 c 4 + abc 3 + ab 3 c + a 3 bc + 1

Walks around dissected polygons 1 α 1 2 7 α 2 3 6 α 3 α 4 4 5 Let D = { α 1 , . . . , α m } be a dissection of a polygon, where the piece α k is a d k -gon, k = 1 , . . . , m .

Walks around dissected polygons 1 α 1 2 7 α 2 3 6 α 3 α 4 4 5 Let D = { α 1 , . . . , α m } be a dissection of a polygon, where the piece α k is a d k -gon, k = 1 , . . . , m . A (counterclockwise) walk from vertex i to vertex j is a sequence of pieces s = ( p i + 1 , p i + 2 , . . . , p j − 1 ) such that (i) p k is incident to vertex k , and (ii) α r appears at most d r − 2 times in s , for any r .

The weight matrix (without edge weights) Let D = { α 1 , . . . , α m } be a dissection of an n -gon. Weight of a piece α k : w ( α k ) = x k ∈ Z [ x 1 , . . . , x m ] = Z [ x ] . Weight of a walk s = ( p i + 1 , . . . , p j − 1 ) : j − 1 � x s = w ( p k ) ∈ Z [ x ] . k = i + 1 For vertices i and j we set � x s ∈ Z [ x ] . w i , j = s : walk from i to j Weight matrix associated to D : W D ( x ) = ( w i , j ) 1 ≤ i , j ≤ n .

1 2 α 7 β 3 6 γ δ 4 5 ab 2 cd  0 1 a + b ab + ac ( a + b )( b + c ) d ( a + b ) bcd  + b 2 + bc +( a + b ) bc ab 2 cd b 2 cd b + c b ( c + d ) + cd 0 1 bcd     ab 2 cd ( a + b ) bcd c + d 0 1 cd bcd     ab 2 cd ( a + b )( b + c ) d ab ( b + c ) d 0 1 d ( b + c ) d     ab 2 + abc ab 2 ( c + d ) ab 2 cd ab + ac + ad 0 1 b + c + d   + b 2 + bc + bd + abd    ab 2 ab 2 c ab 2 cd  a + b ab 0 1 ab ( b + c ) d + ab 2 c ab 2 cd 1 a ab ab ( b + c ) 0 The weight matrix W D is not symmetric!

Complementary symmetry Let D be a polygon dissection with pieces of degree d 1 , . . . , d m . Define a complementing map φ D on weights by giving it on walk x s = � m i = 1 x s i weights (and linear extension): i � m x d i − 2 − s i φ D ( x s ) = . i i = 1 Theorem Let W D = ( w i , j ) be the weight matrix associated to D . Then w j , i = φ D ( w i , j ) for all i , j , i.e., W D is a complementary symmetric matrix.

The determinant of the weight matrix 1 2 α 7 β 3 6 γ δ 4 5 det W D 1 + a 5 b 10 c 3 d 3 + a 2 b 8 c 2 d 2 + a 5 b 8 c 3 d 5 + a 2 b 6 c 2 d 4 = + a 6 b 12 c 4 d 6 + a 3 b 10 c 3 d 5 + a 4 b 6 c 2 d 2 + ab 2 c 3 d + a 5 b 8 c 5 d 3 + a 2 b 6 c 4 d 2 + a 6 b 12 c 6 d 4 + a 3 b 10 c 5 d 3 + a 5 b 6 c 5 d 5 + a 2 b 4 c 4 d 4 + a 6 b 10 c 6 d 6 + a 3 b 8 c 5 d 5 + a 4 b 12 c 6 d 6 + a 7 b 14 c 7 d 7 + a 4 b 4 c 4 d 2 + ab 4 cd + a 3 b 2 cd + a 4 b 4 c 2 d 4 + ab 2 cd 3