On the algebra associated with a geometric lattice

Let L be a geometric lattice. Following P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math. 56 (1980), 167–189, we associate with L a graded commutative algebra A(L). In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including ψ, coincide.     

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

Comparing Codimension and Absolute Length in Complex Reflection Groups

Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. Motivated by connections to the algebraic structure of cohomology governing deformations of skew group algebras, we show that Coxeter groups and the infinite family G(m, 1, n) are the only irreducible complex reflection groups for which reflection length and codimension coincide. We then discuss implications for the degrees of generators of Hochschild cohomology. Along the way, we describe the codimension atoms for the infinite family G(m, p, n), give algorithms using character theory, and determine two-variable Poincaré polynomials recording reflection length and codimension.     

Lattices of Order-Convex Sets of Forests

We give a characterization of the class Co(F)\mathbf{Co}(\mathcal{F}) [Co(F_n)\mathrm{\mathbf{Co}}(\mathcal{F}_n), n < ω, respectively] of lattices isomorphic to convexity lattices of posets which are forests [forests of length at most n, respectively], as well as of the class Co(L)\mathbf{Co}(\mathcal{L}) of lattices isomorphic to convexity lattices of linearly ordered posets. This characterization yields that the class of finite members from Co(F)\mathbf{Co}(\mathcal{F}) [from Co(F_n)\mathbf{Co}(\mathcal{F}_n), n < ω, or from Co(L)\mathbf{Co}(\mathcal{L})] is finitely axiomatizable within the class of finite lattices.     

Finite group actions and étale cohomology

If a finite group G acts on a quasi-projective variety X, then H*_c(X,Z/n), the étale cohomology with compact support of X with coefficients inZ/n, has aZ/n[G]-module structure. It is well known that there is a finer invariant, an object RΓ_c(X,Z/n) of the derived category ofZ/n[G]-modules, whose cohomology is H*_c(X,Z/n). We show that there is a finer invariant still, a bounded complex Λ_c(X,Z/n) of direct summands of permutationZ/n[G]-modules, well-defined up to chain homotopy equivalence, which is isomorphic to RΓ_c(X,Z/n) in the derived category. This complex has many properties analogous to those of the simplicial chain complex of a simplicial complex with a group action. There are similar results forl-adic cohomology.     

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

The geometric structure and the p-rank of an affine triple system derived from a nonassociative moufang loop with the maximum associative center

H. P. Young showed that there is a one-to-one correspondence between affine triple systems (or Hall triple systems) and exp. 3-Moufang loops (ML). Recently, L. Beneteau showed that (i) for any non-associative exp. 3-ML (E, · ) with E = 3ⁿ, 3 Z(E) 3ⁿ⁻³, where n 4 and Z(E) is an associative center of (E, ·), and (ii) there exists exactly one exp. 3-ML, denoted by (E_n, ·), such that E_n = 3ⁿ and Z(E_n) = 3ⁿ⁻³ for any integer n 4. The purpose of this paper is to investigate the geometric structure of the affine triple system derived from the exp. 3-ML(E_n, ·) in detail and to compare with the structure of an affine geometry AG(n, 3). We shall obtain (a) a necessary and sufficient condition for three lines L₁, L₂ and L₃ in (E_n, ·) that the transitivity of the parallelism holds for given three lines L₁, L₂ and L₃ in (E_n, ·) such that L₁L₂ and L₂L₃ and (b) a necessary and sufficient condition for m + 1 points in E_n (1 m < n) so that the subsystem generated by those m + 1 points consists of 3^m points. Using the structure of hyperplanes in (E_n, ·), the p-rank of the incidence matrix of the affine triple system derived from the exp. 3-ML(E_n, ·) is given.     

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that , for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L^′. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Extn, for any integer n?0, using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L=L(λ), even allowing λ to vary, provided the p-adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.     

Embedding and unsolvability theorems for modular lattices

LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ? be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ?(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω. THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ? ?, such thatL(R, ω) is inV. THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ^*) having five generators, where δ^*=δ for infinite δ and δ^*=4δ(m+1) if δ is finite andL has a set ofm generators. THEOREM. The free ?(R)-lattice on ω generators is embeddable in the free ?(R)-lattice on five generators. THEOREM. IfL has an (m, n), ?(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ?(R)-presentation.     

Farrell cohomology of GL(n,Z)

The Tate-Farrell cohomology of GL(n,Z) with coefficients inZ/p is computed forp an odd prime andp−1 ≦n ≦ 2p−3. Its size depends on the Galois structure of the class group of the cyclotomic fieldQ(p√1) and is shown to be quite large in general. Research partially supported by NSF Grant No. DMS-8701758.     

Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r − 1,q ) for r & 3 is the only rank- r(combinatorial) geometry with (q^r − 1) / (q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.     

Weights of the l-adic cohomology of a proper toric variety

We describe the weights of the l-adic cohomology groups of a toric variety over a finite field k in terms of the Ishida complexes of Z-modules. As a consequence, we conclude that, for an r-dimensional proper toric variety X, the m-th cohomology group H^m (X?_k[kbar],Q_l) is of pure weight if m = 0,1,2,3,2r - 3,2r - 2,2r - 1,2r.Furthermore, we show that, for any m such that 3 < m < 2r - 3,there exists an r-dimensional proper toric variety whose m-th cohomology group H^m (X?_k[kbar],Q _l ) is not pure.     

On indecomposable definite unimodular hermitian forms

For any natural numbersm andn≥17 we can construct explicitly indecomposable definite unimodular normal Hermitian lattices of rankn over the ring of algebraic integersR _m in an imaginary quadratic field . It is proved that for anyn (in casem=11, there is one exceptionn=3) there exist indecomposable definite unimodular normal HermitianR ₁₅(R ₁₁)-lattices of rankn, and we exhibit representatives for each class. In the exceptional case there are no lattices with the desired properties. The method given in this paper can solve completely the problem of constructing indecomposable definite unimodular normal HermitianR _m -lattices of any rankn for eachm. Dedicated to the memory of Prof. Lee Hwa-Chung.     

Techniques of computations of Dolbeault cohomology of solvmanifolds

We consider semi-direct products ${\mathbb{C}^{n}\ltimes_{\phi}N}$ of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product ${\mathbb{C}^{n}\times N}$ . As a corollary of this computation, we can compute the Dolbeault cohomology H ^p,q (G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.     

On the cohomology of orbit space of free ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-actions on lens spaces

Let G = ℤ _p , p an odd prime, act freely on a finite-dimensional CW-complex X with mod p cohomology isomorphic to that of a lens space L ^2m−1(p; q ₁, …, q _m ). In this paper, we determine the mod p cohomology ring of the orbit space X/G, when p ² ∤ m.     

The Combinatorics of a Class of Representation Functions

In the study of the irreducible representations of the unitary groupU(n), one encounters a class of polynomials defined onn²indeterminatesz_ij, 1i, jn, which may be arranged into ann×nmatrix arrayZ=(z_ij). These polynomials are indexed by double Gelfand patterns, or equivalently, by pairs of column strict Young tableaux of the same shape. Using the double labeling property, one may define a square matrixD(Z), whose elements are the double-indexed polynomials. These matrices possess the remarkable “group multiplication property”D(XY)=D(X) D(Y) for arbitrary matricesXandY, even though these matrices may be singular. ForZ=UU(n), these matrices give irreducible unitary representations ofU(n). These results are known, but not always fully proved from the extensive physics literature on representation of the unitary groups, where they are often formulated in terms of the boson calculus, and the multiplication property is unrecognized. The generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics. The combinatorial structure of the general polynomials is expected to be intricate, and in this paper, we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums. These special polynomials are denoted byL_{α, β}(Z), whereαandβare integral vectors representing the row sums and column sums of a class of integral matrices. We present a combinatorial interpretation of the multiplicative properties of these polynomials. We also point out the connections with MacMahon's Master Theorem and Schwinger's inner product formula, which is essentially equivalent to MacMahon's Master Theorem. Finally, we give a formula for the double Pfaffian, which is crucial in the studies of the generating function of the 3n−jcoefficients in angular momentum theory. We also review the background of the general polynomials and give some of their properties.     

Cohomological growth rates and Kazhdan-Lusztig polynomials

In a previous work, the authors established various bounds for the dimensions of degree n cohomology and Ext-groups, for irreducible modules of semisimple algebraic groups G (in positive characteristic p) and (Lusztig) quantum groups U _ζ (at roots of unity ζ). These bounds depend only on the root system, and not on the characteristic p or the size of the root of unity ζ. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system ϕ. For quantum groups U _ζ with a fixed ϕ, we show the sequence {max _{L irred} dim H ⁿ (U _ζ , L)} _n has polynomial growth independent of ζ. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that {log max _{L irred} dim H ⁿ (G,L)} has polynomial growth ≤ 3 for any fixed prime p (and ≤ 4 if p is allowed to vary with n). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov and Lubotzky [13].     

Lattices Generated by Orbits of Flats Under Finite Affine Groups

Let AG(n, _q ) be the n-dimensional affine space over  _q . For a given integer m with 0 ≤ m ≤ n, all m-flats form an orbit, denoted by ?(m,n), under the action of the affine group AGL _n ( _q ) of AG(n, _q ). Denote the set of all intersections of m-flats in ?(m,n) by ?(m,n). By ordering ?(m,n) by ordinary or reverse inclusion, two classes of lattices are obtained. This article discusses their geometricity, and computes their character polynomials.     

Syzygies and Diagonal Resolutions for Dihedral Groups

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ω_r(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the “co-represention theorem” H^r(G, N) = Hom_𝒟er(Ω_r(Z), N). We describe the Ω_r(Z) explicitly for the dihedral groups D_4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.