A'Campo--Gusein-Zade diagrams as partially ordered sets

G. G. Ilyuta

Izvestiya. Mathematics , Tome 65 (2001) no. 4, pp. 687-704.

Résumé

The real analogues of many results about complex monodromies of singularities can be formulated and proved in terms of partial orderings on A'Campo–Gusein-Zade diagrams, the real versions of Coxeter–Dynkin diagrams of singularities. In this paper it is proved that the only diagrams among the A'Campo–Gusein-Zade diagrams of singularities that determine partially ordered sets of finite type (in the sense of representations of a quiver) are the diagrams of simple singularities. To encode the real decompositions of a singularity the analogue of Vasilev invariants turn out to be surjections of a partially ordered set onto a chain. Formulae are proved for Arnold $(\operatorname{mod}2)$-invariants of plane curves in terms of the corresponding A'Campo–Gusein-Zade diagrams. We define, in the context of higher Bruhat orders, higher partially ordered sets and we describe their connection with the higher $M$-Morsifications $A_n$. We also consider certain previously known results about real singularities from the point of view of partially ordered sets.

Bibliographie

[1] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii, T. II, Nauka, M., 1984 | MR

[2] Arnold V. I., “Ischislenie zmei i kombinatorika chisel Bernulli, Eilera i Springera grupp Kokstera”, UMN, 47:1 (1992), 3–45 | MR | Zbl

[3] Gabrielov A. M., “Diagrammy Dynkina unimodalnykh osobennostei”, Funkts. analiz i ego prilozh., 8:3 (1974), 1–6 | MR | Zbl

[4] Gusein-Zade S. M., “Matritsy peresechenii dlya nekotorykh osobennostei funktsii dvukh peremennykh”, Funkts. analiz i ego prilozh., 8:1 (1974), 11–15 | MR | Zbl

[5] Gusein-Zade S. M., “Indeks osoboi tochki gradientnogo vektornogo polya”, Funkts. analiz i ego prilozh., 18:1 (1984), 7–12 | MR | Zbl

[6] Gusein-Zade S. M., “Diagrammy Dynkina nekotorykh polnykh peresechenii i veschestvennye morsifikatsii”, Tr. MIRAN, 209 (1995), 124–132 | MR | Zbl

[7] Ilyuta G. G., “O preobrazovanii Kokstera izolirovannoi osobennosti”, UMN, 42:2 (1987), 227–228 | MR | Zbl

[8] Ilyuta G. G., “Kharakterizatsiya veschestvennykh diagramm Kokstera–Dynkina prostykh osobennostei”, UMN, 50:3 (1995), 157–158 | MR | Zbl

[9] Ilyuta G. G., “Vysshie poryadki Bryua i $M$-morsifikatsii”, Dokl. RAN, 347:3 (1996), 306–308 | MR | Zbl

[10] Ilyuta G. G., “Geometricheskie realizatsii vysshikh poryadkov Bryua i $M$-morsifikatsii”, Izv. RAN. Ser. matem., 60:6 (1996), 91–100 | MR | Zbl

[11] Ilyuta G. G., “Diagrammy Kokstera–Dynkina chastichno uporyadochennykh mnozhestv i diagrammy A'Kampo–Gusein-Zade prostykh osobennostei”, Funkts. analiz i ego prilozh., 31:1 (1997), 12–19 | MR | Zbl

[12] Ilyuta G. G., “Sootnosheniya iz teorii Morsa dlya diagramm Kokstera–Dynkina”, Funkts. analiz i ego prilozh., 34:3 (2000), 73–77 | MR | Zbl

[13] Lyashko O. V., “Raspadenie prostykh osobennostei funktsii”, Funkts. analiz i ego prilozh., {10}:2 (1976), 49–56 | MR | Zbl

[14] Stenli R., Perechislitelnaya kombinatorika, Mir, M., 1990 | MR

[15] Chislenko Yu. S., “Raspadeniya prostykh osobennostei veschestvennykh funktsii”, Funkts. analiz i ego prilozh., 22:4 (1988), 52–67 | MR

[16] A'Campo N., “Le groupe de monodromie du deploiment des singularites isolees de courbes planes, I”, Math. Ann., 213 (1975), 1–32 | DOI | MR | Zbl

[17] A'Campo N., “Le groupe de monodromie du deploiment des singularites isolees de courbes planes, II”, Actes du Congr. Internat. Math., V. 1 (Vancouver), 1974, 395–404 | MR

[18] A'Campo N., “Real deformations and complex topology of plane curve singularities”, Ann. Faculte Sci. Toulouse, VIII (1999), 5–23 | MR

[19] A'Campo N., Generic immersions of curves, knots, monodromy and gordian number, Preprint, Univ. Basel, 1999

[20] A'Campo N., Planar trees, slalom curves and hyperbolic knots, Preprint, Univ. Basel, 1999

[21] Aicardi F., “Tree-like curves”, Adv. Soviet Math., 21 (1994), 1–31 | MR | Zbl

[22] Arnold V. I., “Bernoulli–Euler updown numbers, associated with function singularities, their combinatoric and arithmetic”, Duke Math. J., 63 (1991), 537–555 | DOI | MR | Zbl

[23] Arnold V. I., “Springer numbers and morsification spaces”, J. Alg. Geom., 1 (1992), 197–214 | MR | Zbl

[24] Arnold V. I., Topological invariants of plane curves and caustics, University Lecture Series, 5, 1994, Providence, RI | MR

[25] Arnold V. I., “Topological classification of real trigonometric polynomials and cyclic serpents plyhedron”, Arnold–Gelfand Math. Seminars, Birkhauser, Boston, 1997, 79–82 | MR

[26] Artin M., Verdier J., “Reflexive modules over rational double points”, Math. Ann., 270 (1985), 79–82 | DOI | MR | Zbl

[27] Auslander M., “Rational singularities and almost split sequences”, Transl. Amer. Math. Soc., 293 (1986), 511–531 | DOI | MR | Zbl

[28] Baclawsky K., “Galois connections and the Leray spectral sequence”, Adv. Math., 25 (1977), 191–215 | DOI | MR

[29] Baclawsky K., “The Möbius algebra as a Grothendieck ring”, J. Algebra, 57 (1979), 167–179 | DOI | MR

[30] Bjorner A., Wachs M., “Permutation statistics and linear extensions of posets”, J. Comb. Th. Ser. A, 58 (1991), 85–114 | DOI | MR

[31] Bjorner A., Wachs M., “Shellable nonpure complexes and posets, I”, Trans. Amer. Math. Soc., 348 (1996), 1299–1327 | DOI | MR

[32] Brenner S., Butler M., “Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors”, Lecture Notes in Math., 832 (1980), 103–169 | DOI | MR | Zbl

[33] Brightwell G., “Balanced pairs in partial orders”, Discrete Math., 201 (1999), 25–52 | DOI | MR | Zbl

[34] Daykin D., Daykin J., Paterson S., “On log-concavity for order-preserving maps of partial orders”, Discrete Math., 50 (1984), 221–226 | DOI | MR | Zbl

[35] Daykin J., “Inequalities for the number of monotonic functions of partial orders”, Discrete Math., 61 (1986), 41–55 | DOI | MR | Zbl

[36] Ebeling W., “Milnor lattices and geometric bases of some special singularities”, Enseign Math., 29 (1983), 263–280 | MR | Zbl

[37] Ebeling W., Wall C. T. C., “Kodaira singularities and an extension of Arnold's strange duality”, Comp. Math., 56 (1985), 3–77 | MR | Zbl

[38] Ebeling W., Gusein-Zade S., “Coxter–Dynkin diagrams of the complete intersection singularities $Z_9$ and $Z_{10}$”, Math. Z., 218 (1995), 549–562 | DOI | MR | Zbl

[39] Ebeling W., Gusein-Zade S., “Coxeter–Dynkin diagrams of fat points in $\mathbb{C}^2$ and their stabilizations”, Math. Ann., 306 (1996), 487–512 | DOI | MR | Zbl

[40] Edelman P., Hibi T., Stanley R., “A recurrence for linear extensions”, Order, 6 (1989), 15–18 | DOI | MR | Zbl

[41] Ehrenborg R., “On posets and Hopf algebras”, Adv. Math., 119 (1996), 1–25 | DOI | MR | Zbl

[42] Etienne G., “Linear extensions of finite posets and a conjecture of G. Kreweras on permutations”, Discrete Math., 52 (1984), 107–112 | DOI | MR

[43] Fishburn P., “Correlation in partially ordered sets”, Discrete Appl. Math., 39 (1992), 173–191 | DOI | MR | Zbl

[44] Ford D., McKay J., “Representations and Coxeter graphs”, The Geometric Vein (The Coxeter Festschrift), ed. Ch. Davis, B. Grunbaum, F. Sherk, Springer-Verlag, Berlin, 1981, 549–554 | MR

[45] Gonzales-Sprinberg G., Verdier J., “Construction geometrique de la correspondance de McKay”, Ann. Sci. E.N.S., 16 (1983), 409–449 | MR

[46] Goryunov V., “Subprincipal Springer cones and morsifications of Laurent polynomials and $D_\mu$ singularities”, Adv. in Soviet. Math., 21 (1994), 163–188 | MR | Zbl

[47] Goryunov V., “Morsifications of rational functions”, AMS Translations, 173 (1996), 85–96 | MR | Zbl

[48] Goryunov V., Baines C., Möbius and odd real trigonometric $M$-functions, Preprint, 1997 | MR

[49] Gusein-Zade S., “Invariants of generic plane curves and invariants of singularities”, Proc. Symp. Pure Math., 62:1 (1997), 423–433 | MR | Zbl

[50] Gusein-Zade S., Natanzon S., “The Arf-invariant and the Arnold invariants of plane curves”, Arnold–Gelfand Math. Seminars, Birkhauser, Boston, 1997, 267–280 | MR | Zbl

[51] Hibi T., “Toric varieties arising from canonical triangulations of poset polytopes are projective”, Words, languages, and combinatorics, V. II (Kyoto, 1992), World Sci. Publ, 1994, 192–199 | MR | Zbl

[52] Kahn J., Saks M., “Balancing poset extensions”, Order, 1 (1984), 113–126 | DOI | MR | Zbl

[53] Kahn J., Linual N., “Balancing extensions via Brunn–Minkovski”, Combinatorica, 11 (1991), 363–368 | DOI | MR | Zbl

[54] Kierstead H., Trotter W., “Super-greedy linear extensions of ordered sets”, Ann. New York Acad. Sci., 555 (1989), 262–271 | DOI | MR | Zbl

[55] MacPherson R., “The combinatorial formula of Gabrielov, Gelfand, and Losik for the first Pontrjagin class”, Sem. Bourbaki, no. 497, 1976/77; Lecture Notes in Math., 677, 1978 | MR | Zbl

[56] Manin Yu., Schechtman V., “Arrangements of hyperplanes, higher braid groups and higher Bruhat orders”, Adv. Studies in Pure Math., 44 (1985), 289–308 | MR

[57] Pruesse G., Ruskey F., “Generating the linear extensions of certain posets by transpositions”, SIAM J. Discrete Math., 4 (1991), 413–422 | DOI | MR | Zbl

[58] Ray N., Schmitt W., “Combinatorial models for coalgebraic structures”, Adv. Math., 138 (1998), 211–262 | DOI | MR | Zbl

[59] Ruskey F., “Transposition generation of alternating permutations”, Order, 6 (1989), 227–233 | DOI | MR | Zbl

[60] Ruskey F., “Generating linear extensions of posets by transpositions”, J. Comb. Th. Ser. B, 54 (1992), 77–101 | DOI | MR | Zbl

[61] Rival I., Zagulia N., “Greedy linear extensions with constraints”, Discrete Math., 63 (1987), 249–260 | DOI | MR | Zbl

[62] Stanley R., Ordered structures and partitions, Mem. Amer. Math. Soc., 119, 1972 | MR | Zbl

[63] Stanley R., “Two combinatorial applications of the Aleksandrov–Fenchel inequalties”, J. Comb.Th. Ser. A, 31 (1981), 56–65 | DOI | MR | Zbl

[64] Stanley R., “Two poset polytopes”, Discrete Comput. Geom., 1 (1986), 9–23 | DOI | MR | Zbl

[65] Steiner G., “Minimizing bumps in ordered sets by substitution decomposition”, Discrete Math., 76 (1989), 285–289 | DOI | MR | Zbl

[66] Trotter W., Combinatorics and partially ordered sets. Dimension theory, Johns Hopkins Univ. Press, Baltimore, 1992 | MR | Zbl

[67] Trotter W., “Partially ordered sets”, Handbook of Combinatorics, Ch. 8, Elsevier Sci., Amsterdam, 1995 | MR | Zbl

[68] Wagner D., “Enumeration of functions from posets to chains”, European J. Comb., 13 (1992), 313–324 | DOI | MR | Zbl

[69] Yuzvinsky S., “Linear representations of posets, their cohomology and a bilinear form”, European J. Comb., 2 (1981), 385–397 | MR | Zbl

[70] Yuzvinsky S., “The variety of primitive representations of a quiver with normal relations”, Commun. Alg., 11 (1983), 973–1009 | MR | Zbl

[71] Yuzvinsky S., “Flasque sheaves on posets and Cohen–Macaulay unians of regular varieties”, Adv. Math., 73 (1989), 24–42 | DOI | MR | Zbl

[72] Yuzvinsky S., “Projective modules over rings of global sections of flasque sheaves”, J. Pure Appl. Alg., 65 (1990), 191–197 | DOI | MR | Zbl

[73] Ziegler G., “Higher Bruhat orders and cyclic hyperplane arrangements”, Topology, 32 (1993), 259–279 | DOI | MR | Zbl