Двумерные клеточные комплексы, вещественные числа Гурвица и интегрируемые системы тема диссертации и автореферата по ВАК РФ 00.00.00, доктор наук Бурман Юрий Михайлович

• Бурман Юрий Михайлович
• доктор наук
• 2023

Оглавление диссертации доктор наук Бурман Юрий Михайлович

Contents

Introduction

1. General theory and Hurwitz numbers

1.1. Real (twisted) case

1.1.1. A topological definition: ribbon decompositions

1.1.2. An algebraic model: symmetric sequences of transpositions

1.1.3. An algebro-geometric model: twisted branched coverings

1.2. Relation between the models and the generating function

1.2.1. Equivalence of the definitions

1.2.2. Twisted cut-and-join equation

1.3. Complex (oriented) case and variants of the theory

1.3.1. Geometry of ribbon decompositions

1.3.2. Oriented ribbon decompositions

1.3.3. Oriented (complex) Hurwitz numbers

2. Low genus case

2.1. Functions of rank 1 operators 8 2.1.1. Discrete path integration 8 Corollaries: discrete differential geometry and matrix-tree theorems

2.2. Variants of the theory

2.2.1. PfafRans

2.2.2. Lie elements

3. Miscellaneous

3.1. Directed graphs and higher determinants

3.1.1. Matrix-tree theorem for Bernardi polynomials

3.1.2. Higher determinants

3.2. Graphs with a given core and /-determinants

3.2.1. Matrix-subgraph theorem 16 /

3.2.3. Matroids and the matrix-tree theorem for the root system D

3.3. Ramified coverings for curves

3.3.1. Ramified coverings in Severi varieties

3.3.2. Ramified coverings in dimension

20

Article 1. Triangulation of surfaces with boundary and the

homotopy principle for functions without critical points

Article 2. Around matrix-tree theorem 36 Article 3. Quasiharmonic polynomials for Coxeter groups

and representations of Cherednik algebras

1

Article 5. Matrix-tree theorems and discrete path integration 101 Article 6. On projections of smooth and nodal plane curves 121 Article 7. Higher matrix-tree theorems and Bernardi

polynomial

Article 8. On Hurwitz-Severi numbers 161 Article 9. Algebraic relations between harmonic and

anti-harmonic moments of plane polygons

Article 10. Lie elements and the Matrix-tree theorem

References

Appendix A.

Appendix B.

Appendix c.

Appendix D.

Appendix E.

Appendix F.

Appendix G.

Appendix H.

Appendix I.

Appendix J.

Введение диссертации (часть автореферата) на тему «Двумерные клеточные комплексы, вещественные числа Гурвица и интегрируемые системы»

Introduction I. General theory and Hurwitz numbers

Most theorems described in this manuscript deal with description of geometric objects (mostly holomorphic mappings between curves) with prescribed singularities, as well as to the combinatorics necessary to do it. In this section we relate topological objects (ribbon decompositions) to meromorphic functions on curves with prescribed critical values. Curves either come as they are or with some additional structure (like real one).

1.1. Real (twisted) case. In this section we describe ribbon decompositions on real curves without real points, that is, complex curves equipped with an anti-holomorphic involution having no fixed points.

1.1.1. A topological definition: ribbon decompositions. Fix a positive integer m and a partition (Ai > • • • > As) of the number n = |A| = Ai + • • • + As into s parts. The main object of study in this section, the twisted Hurwitz numbers hm have several definitions or models, as we call them.

The first, topological model, uses ribbon decompositions.

Definition 1.1.1. Decorated-boundary surface (DBS) is a triple (M, (a1;..., an), (o1,..., on)) where M is a compact surface (real 2-manifold) with boundary, a1,..., an e dM are marked points and every o» is a local orientation of dM (hence, of M itself, too) in the vicinity of the point a», such that

• every connected component of M has nonempty boundary, and

• every connected component of dM contains at least one point a».

The DBS M and M' with the same number n of marked points are called equivalent if there exists a homeomorphism h : M ^ M' such that h(a{) = aj and h*(oi) = oj for all i = 1,..., n. The set of equivalence classes of DBS with n marked points will be denoted DBSn.

Pick marked points aj,aj e dM, and let £j; £j e {+, —}. Consider points aj, aj e dM lying near a», aj and such that the boundary segment ajaj is directed along the orientation o» if e» = + and opposite to it if e» = —; the same for j. Now take a long narrow rectangle with a distinguished diagonal ("a ribbon" henceforth) and glue its short sides to dM so that the endpoints of the diagonal are identified with a» and aj and the other two vertic es, with a^d aj. The result of gluing is homeomorphic to a surface M' with the boundary dM' 9 a1;..., an. The boundary of M' near a^d aj contains a segment of dM (the "old" part) and a segment of a long side of the ribbon glued (the "new" part); define local orientations o», oj of dM' near a», aj so that the orientations of the "old" parts were preserved; for k = i,j take o'k = ok by definition. Now (M', (a1;..., an), (o1,..., on)) is a DBS, so we defined a mapping G[i,j]ei: DBSn ^ DBSn called ribbon gluing. The ribbon gluing G[i,j]£i'£j be called twisted if e» = ej, and non-twisted otherwise.

Denote by En e DBSn a union of n disks with one marked point on the

M

M = G[im, jm]£m. . . G[i1, j1]£1 A En.

Definition 1.1.2. The twisted Hurwitz number is defined as h~ A =f where &m,\ is ^^e set of decompositions into m ribbons of surfaces having s boundary components containing A1;..., As marked points.

1.1.2. An algebraic model: symmetric sequences of transpositions. Another model for hm A is algebraic. Consider a fixed-point-free involution t = (1,n + 1)(2,n + 2)... (n, 2n) in the symmetric group £2«• Its centralizer C(t) is isomorphic to the Coxeter group Bn with the reflect ions rij = (ij )(j) and lij = (ii); here 1 < i < j < 2n and i =f i + n mod 2n.

Denote by C~(t) = {a G S2n | Ta = a-1T} (a "twisted centralizer" of t). Let a = c1... cm G C^(t ) wher e c1,... ,cm are independent cycles. It is easy to see that for every i either there exists j = i such that ci = (u1.. . uk) and j = (t(uk)... t(u^), or ci has even length 2k and looks like ci = (u1 ... ukt(uk)... t(u1)) In the first case we say that the cycles c^d cj are T-symmetric, and in the second case the cycle c^ is T-self-symmetric.

Theorem 1.1.3. fl] There exists a one-to-one correspondence between the following three sets:

1. The quotient (the set of left cosets) S2n/Bn where we assume Bn is a centralizer of t;

2. The set B^ of permutations a G C^(t) such that their cycle decomposition contains no T-self-symmetric cycles.

3. The set of fixed-point-free involutions A G S2n.

The size of each set is (2n - 1)!! = 1 x 3 x • • • x (2n - 1).

The correspondences are a ^ ara-1T (cosets to twisted centralizer), a ^ aT (centralizer to involutions), a ^ X where X is the set of all x such that a = xtx-1 (involutions to cosets).

Let A = (A1 > • • • > As) be a partition of n. Denote by C^(t) c C~(t) the set of permutations whose decomposition into independent cycles consists of s pairs of T-symmetric cycles of lengths A1;..., As.

Definition 1.1.4. The twisted Hurwitz number is defined as h\"m A =f a-

Here Hm,A is ^^e set of sequences (a1,... ,am^f m transpositions such that the conjugacy class containing the permutation a1... am is mapped to an element of c^(t) by the correspondence of Theorem 1.1.3.

1.1.3. An algebro-geometric model: twisted branched coverings. Let TV be a closed real algebraic curve without real points, i.e. a compact complex curve with an anti-holomorphic involution T having no fixed points. The quotient of TV by the action of T will be denoted N; it is a closed surface (compact 2-manifold without boundary), not necessarily orientable. Denote by p : TV ^ N the quotient map.

Denote by M = CP1/{z ~z) = iu{ ro} where H c C is the upper half-plane; its closure H is homeomorphic to a disk. Let 7r : CP1 —> H be the quotient map. The following definition belongs to G. Chapuy and M. Dol§ga:

Definition 1.1.5. [11] A continuous map / : N —> H is called a twisted branched covering if there exists a a real holomorphic map / : TV ^ CP1 (in the classical sense, a T-invariant holomorphic map) such that n o / = / o p, and all the critical

/V

This definition implies in particular that the ramification profile of any critical value c G RP1 C CP1 of / has every part repeated twice: (A1; A1;..., As, As), and

deg / = 2n is even. In this case we say that the ramification profile of the critical value 7r(c) € <9H of the map / : N —> H is A = (Ai,..., As).

Twisted branched coverings are split into equivalence classes via right-left equivalence.

Definition 1.1.6. The twisted Hurwitz number is defined as h~ A =f 2^j#®m,A-Here Dm,A is the set of equivalence classes of twisted branched coverings having m critical values with the ramification profiles 211n-2 and one critical value to with the ramification profile A.

1.2. Relation between the models and the generating function. In the

Section 1.1 we described several models (finite sets) describing real functions with real critical values of prescribed structure. Here we relate these models and study-generating function of their sizes.

1.2.1. Equivalence of the definitions.

Theorem 1.2.1. The three definitions of the twisted Hurwitz numbers are equivalent: = #@m, A = #*5m, A-

The second equality is proved in [1] by a direct correspondence S between the sets Hm,A- To prove the first equality we show in [1] that the gener-

ating function of the twisted Hurwitz numbers satisfies a PDE of parabolic type called twisted cut-and-join equation; see Section 1.2.2 below. Cardinalities of the sets Dm,A are shown in [11] to satisfy the same equation with the same initial data. Finding a direct 2m : 1 correspondence between the sets Dm,A and &m,A (or Hm,A) was later found in [12] and also (snother construction) in the paper [13] (in preparation).

1.2.2. Twisted cut-and-join equation. Consider the generating function of the twisted Hurwitz numbers defined as follows:

TTOM = E E ^TP^PX, ■■■pxj™-

z—' z—' m!

m>0 A

Theorem 1.2.2. [1] Hr~J satisfies the cut-and-join equation = CJ~(H~)

where

CJ~ = + + + ^'Hk ~ 1)Pki

d d2 = J2(i + j)(PiPi+Pi+i)o^: + ^

The twisted cut-and-join operator is the member of the Laplace-Beltrami family-corresponding to the parameter a = 2; another member (for a = 1) is the classical cut-and-join [14]. Like all Laplace-Beltrami operators, the twisted cut-and-join has a full system of polynomial eigenfunctions indexed by partitions; in this case they are zonal polynomials ZA (see [15]).

Theorem 1.2.3. [1]

= E^P^EW " 0) ^(2^2) •

where ^a(a) = U{ij)ey(a) (aa(i, j)+l(i, j) + 1) and $A(a) = H{ij)ey(a)(aa(i,j) + l(i,j)+a). HereY (A) is the Young diagram of the partition A, and a(i,j) and l(i,j) are the arm and the leg, respectively, of the cell (i, j) G Y(A).

1.3. Complex (oriented) case and variants of the theory. Here we apply the technique of ribbon decompositions to complex curves without real structure. Some models for this case were classically known (and served an inspiration for our work); we relate ribbon decompositions to them.

1.3.1. Geometry of ribbon decompositions. Recall that every ribbon of a ribbon decomposition is supplied with a diagonal; the union of them is a graph embedded into M called the diagaonal graph. Each ribbon also carries a number. The connected components of the complement of the diagonal graph are called faces.

M

diagonal graph; call them external edges and denote the graph obtained by G. Theorem 1.3.1. fl]

1. Vertices, edges and faces of the graph G form a cell decomposition of M (as 0-ce//s, 1-cells and 2-cells, respectively); in particular, every face is homeomorphic to a disk. Every face is adjacent to exactly one external edge. The total number of faces is equal to the number of vertices (i.e. marked points).

2. The diagonal graph r is a homotopy retract of the surface M.

1.3.2. Oriented ribbon decompositions. A ribbon decomposition

M = G[im, jm]£m... G[i1, j1jei'il En

is called oriented if all the signs ei, Si = +. In this case M has a natural orientation and all the local orientations are consistent with it. M

boundary according to the orientation form a decomposition of some permutation a G Sn into independent cycles; a is called the boundary permutation of M

Theorem 1.3.2. [6, 1] The diagonal graph r of an oriented, ribbon decomposition has the following properties:

1. (vertex monotonicity) For every vertex ai o/ r the sequence of numbers of

ai

I1 < ••• <4.

2. (face monotonicity) For every face /j, of r the numbers l1;...,lp of the internal edges adjacent to it are increasing if the count starts at the (only)

/i

3. (face separation) Every internal edge ofT separates two different faces.

4. (boundary permutation) Letaik and ajfc be endpoints ofthe edge ek of r, k = 1,..., m. Then the boundary permutation of M is equal to (imjm)... (i1j1) G

Sn

1.3.3. Oriented (complex) Hurwitz numbers.

Definition 1.3.3. The (classical) Hurwitz number is defined as hmt\ =f A

where &m A is the set of oriented decompositions into m ribbons of surfaces having s boundary components containing A1;..., As marked points.

Definition 1.3.4. The Hurwitz number is defined as hmt\ =f Here ¿5™ A

is the set of sequences (a1,..., am) of m transpositions that belong to the set Hm,A and such that every k one has ak = (ik, jk) where 1 < ik < jk < n. Equivalently, Hm a can defined as a set of sequences (a1;..., am) of m transpositions in Sn such that their product has the cyclic structure defined by the partition A (one cycle of length A1; one of A2, etc.)

Definition 1.3.5. The Hurwitz number is defined as hmt\ =f ^hii^^m Here Dm A is the set of equivalence classes of twisted branched coverings from the set Dm,A where the curve N consists of two copies of the curve N exchanged by the involution T. Equivalently, hm<\ can be defined as ^#2?m,A where Vmt\ is the set of meromorphic functions f : N ^ CP1 having m critical values with the ramification profiles 211n-2 ^^d one critical value to with the ramification profile A

Theorem 1.3.6. The three definitions of the classical Hurwitz numbers are equivalent: #Dm,x = #sm,A = #<a-

The second equality is classically known (see e.g. [14] and the references therein); the first one is proved in [1].

Besides the oriented case described in this manuscript the theory of Hurwitz numbers has many other versions: for Coxeter groups (R. Fesler, [2]), for arbitrary-ramification profiles (M. Lozhkin, [3]), and more.

Список литературы диссертационного исследования доктор наук Бурман Юрий Михайлович, 2023 год

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