Introduction Results Towards algebraic tools Conclusion A generalization of the quadrangulation relation to constellations and hypermaps Wenjie Fang, LIAFA FPSAC 2013 26 July 2013, Universit´ e Sorbonne Nouvelle - Paris 3 Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Motivation A planar quadrangulation ... Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Motivation ... is always bipartite, ... Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Motivation ... which is not true in higher genus. n m Planar case Case g = 1 (on a torus) (bipartite iff m, n are even) Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Motivation Quadrangulation relation Let Q ( g ) and B ( g,k ) be the number of quadrangulations (resp. bipartite n n quadrangulations with marked vertices) with: n edges, g as genus, k marked black vertices. Theorem (The quadrangulation relation (Jackson and Visentin, 1990)) We have the following relation. Q ( g ) = 2 2 g B ( g, 0) + 2 2 g − 2 B ( g − 1 , 2) + 2 2 g − 4 B ( g − 2 , 4) . . . n n n n For the planar case, we have Q (0) = B (0 , 0) . n n Obtained using algebraic method, can be generalized to general bipartite maps (Jackson and Visentin (1999)). Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Motivation Asymptotic behavior of quadrangulations We admit that the number of bipartite quadragulation of fixed genus g 5 2 ( g − 1) 12 n ) . ( c.f. Bender and Canfield (1986)) grows as Θ( n Theorem (The quadrangulation relation (Jackson and Visentin, 1990)) We have the following relation. Q ( g ) = 2 2 g B ( g, 0) + 2 2 g − 2 B ( g − 1 , 2) + 2 2 g − 4 B ( g − 2 , 4) . . . n n n n The first term dominates, and we have Q ( g ) ∼ 2 2 g B ( g, 0) . n n Corollary For any fixed g , the probability for a quadrangulation of genus g with n edges to be bipartite converges to 2 − 2 g when n → ∞ . Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Results Constellations and hypermaps The m -constellations can be seen as a generalization of bipartite maps. bipartite map m -constellation 2 colors m colors edges hyperedges (black) faces hyperfaces (white) even degree degree divisible by m C.f. Lando and Zvonkin (2004), also Bousquet-M´ elou and Schaeffer (2000), Bouttier, Di Francesco and Guitter (2004). We define m -hypermaps as the counterpart of ordinary maps for m -constellations, i.e. without coloring. Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Results Our generalization Let H ( g ) n,m and C ( g,l 1 ,...,l m − 1 ) be the number of m -hypermaps (resp. n,m m -constellations with marked vertices) with: n hyperedges, g as genus, l i marked vertices with color i . Theorem (Our generalized relation) We have the following relation: g � � c ( m ) H ( g ) m 2 g − 2 i l 1 ,...,l m − 1 C ( g − i,l 1 ,...,l m − 1 ) n,m = . n,m i =0 l 1 + ... + l m − 1 =2 i Here, the coefficients c ( m ) l 1 ,...,l m − 1 are all positive integers with explicit expression. Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Results Some examples Corollary (Our generalized relation, case m = 2 , 3 , 4 ) g H ( g ) � 2 2 g − 2 i C ( g − i,l, 2 i − l ) n, 2 = , n, 2 i =0 g 2 i 2 · 2 l + ( − 1) l H ( g ) � � C ( g − i,l, 2 i − l ) 3 2 g − 2 i n, 3 = , n, 3 3 i =0 l =0 g 2(3 l 1 2 l 2 + 2 l 2 ( − 1) l 1 ) H ( g ) C ( g − i,l 1 ,l 2 , 2 i − l 1 − l 2 ) � � 4 2 g − 2 i n, 4 = . n, 4 4 i =0 l 1 ,l 2 ≥ 0 l 1 + l 2 ≤ 2 i Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Results Application: asymptotic counting In Chapuy (2009), the number C ( g ) n,m = C ( g, 0 ,..., 0) of m -constellations n,m 5 2 ( g − 1) ρ n behaves as Θ( n m ) when n tends to infinity. We recover the following result given by Chapuy (2009). Corollary (Asymptotique behavior of m -hypermaps) When n tends to infinity, H ( g ) n,m ∼ m 2 g C ( g ) n,m . Our relation can be viewed as a “higher order development” of this corollary. Corollary For any fixed g , the probability for an m -hypermap of genus g with n hyperedges to be an m -constellation converges to m − 2 g when n → ∞ . Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Results Algebraic approach of maps Combinatorial maps Combinatorics (Transitive) rotation systems Decompositions of the identity element in S n Algebra Group algebra, characters, representation theory Example : Goupil and Schaeffer (1998), Goulden and Jackson (2008), Poulalhon and Schaeffer (2002), Goulden, Guay-Paquet and Novak (2012), etc... Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Towards algebraic tools Maps as decompositions Map model Decomposition form Group m -constellation S n σ 1 σ 2 · · · σ m φ = id with n hyperedges (hyperedges) σ • σ ◦ φ = id m -hypermap S mn with σ • of cycle type [ m n ] with n hyperedges (edges) and σ ◦ of cycle type mµ For a partition µ = ( µ 1 , µ 2 , . . . , µ k ) , we note mµ the scaled partition ( mµ 1 , mµ 2 , . . . , mµ k ) . 1 m 3 7 1 φ 2 σ 1 2 5 1 m cycle: (1 , 3 , 7 , 2 , 5) Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Towards algebraic tools Counting decompositions By the general representation theory, the number of decompositions of the form σ 1 σ 2 · · · σ m = id , with σ i of cycle type λ ( i ) , can be expressed with characters evaluated at each λ ( i ) . Frobenius formula The number of such decompositions is � m � m 1 � � � χ θ # C λ ( i ) λ ( i ) . dim ( V θ ) m # S n i =1 i =1 θ ⊢ n Here C λ is the set of permutations with cycle type λ . Then, for m -hypermaps, we need to evaluate characters in S mn at mµ . How to exploit? Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Towards algebraic tools Key algebraic result - factorization of χ θ mµ In fact, we can express a character of the form χ θ mµ of S mn with characters in smaller groups. The following theorem generalizes results in Jackson and Visentin (1990) and in the book of James and Kerber (1981). Theorem (Factorization of certain characters (W.F.)) Let m, n be positive integers, and µ ⊢ n , θ ⊢ mn two partitions. We have m χ θ ( i ) � � µ ( i ) z − 1 χ θ mµ = z µ sgn( π θ π ′ θ ) µ ( i ) . i =1 µ (1) ⊎···⊎ µ ( m ) = µ Here z µ = # S n / # C µ . Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Towards algebraic tools Two possible approaches There are two different approaches to obtain this result. Algebraic approach Using the Jacobi-Trudi identity, we can express χ θ mµ with a determinant, which has a block structure, resulting in the wanted factorization. Combinatorial approach There is a combinatorial interpretation of χ θ mµ using ribbon tableaux. In the framework of the boson-fermion correspondence, it gives the wanted character factorization. Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
Introduction Results Towards algebraic tools Conclusion Towards algebraic tools Algebraic approach via an example : m = 3 We try to evaluate χ θ 3 µ , with θ = (6 , 6 , 4 , 4 , 4 , 3 , 3) . χ θ 3 µ = z 3 µ [ p 3 µ ] s θ Here, p 3 µ is the powersum symmetric funtion indexed by 3 µ , s θ the Schur function indexed by the partition θ , and z λ = # S n / # C λ for λ ⊢ n . This is a consequence of the change of basis from Schur functions to powersum functions in the symmetric function ring. Wenjie Fang, LIAFA A generalization of the quadrangulation relation to constellations and hypermaps
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