Conjectural Description Research Articles

Degenerate affine Grassmannians and loop quivers

We study the connection between the affine degenerate Grassmannians in type A, quiver Grassmannians for one vertex loop quivers, and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type GLn and identify it with semi-infinite orbit closure of type A2n−1. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional appro- ximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type A1(1), propose a conjectural description in the symplectic case, and discuss the generalization to the case of the affine degenerate flag varieties.

Ulrich Schur bundles on flag varieties

In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal ample class. We classify Ulrich bundles of this form on two-step flag varieties F(1,n−1;n), F(2,n−1;n), F(2,n−2;n), F(k,k+1;n) and F(k,k+2;n). We give a conjectural description of the two-step flag varieties which admit such Ulrich bundles.

Rigid inner forms of real and p-adic groups

We dene a new cohomology set H 1 (u ! W;Z ! G) for an ane algebraic group G and a nite central subgroup, both dened over a local eld of characteristic zero, which is an enlargement of the usual rst Galois cohomology set of G. We show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered L-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the case of real groups, we show that this description is correct using Shelstad’s work.

The Orlik-Terao Algebra and the Cohomology of Configuration Space

ABSTRACTWe give a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement of type An − 1 as a graded representation of Sn, and we give a conjectural description of this representation in terms of the cohomology of the configuration space of n points in SU(2) modulo translation. We also give a version of this conjecture for more general graphical arrangements.

Fluctuations of TASEP and LPP with general initial data

We prove Airy process variational formulas for the one-point probability distribution of (discrete time parallel update) TASEP with general initial data, as well as last passage percolation from a general down-right lattice path to a point. We also consider variants of last passage percolation with inhomogeneous parameter geometric weights and provide variational formulas of a similar nature. This proves one aspect of the conjectural description of the renormalization fixed point of the Kardar–Parisi–Zhang universality class.

Extremal divisors on moduli spaces of rational curves with marked points

We study effective divisors on $\overline{M}_{0,n}$, focusing on hypertree divisors introduced by Castravet and Tevelev and the proper transforms of divisors on $\overline{M}_{1,n-2}$ introduced by Chen and Coskun. Results include a database of hypertree divisor classes and closed formulas for Chen--Coskun divisor classes. We relate these two types of divisors, and from this construct extremal divisors on $\overline{M}_{0,n}$ for $n \geq 7$ that furnish counterexamples to the conjectural description of the effective cone of $\overline{M}_{0,n}$ given by Castravet and Tevelev.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math.163(2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

The Strong Suslin Reciprocity Law and Its Applications to Scissor Congruence Theory in Hyperbolic Space

We prove the strong Suslin reciprocity law conjectured by A. B. Goncharov and describe its corollaries for the theory of scissor congruence of polyhedra in hyperbolic space. The proof is based on the study of Goncharov’s conjectural description of certain rational motivic cohomology groups of a field. Our main result is a homotopy invariance theorem for these groups.

Homology of depth-graded motivic Lie algebras and koszulity

The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group of a Lie algebra with quadratic presentation, constructed out of the period polynomials of modular forms; (b) the koszulity of the enveloping algebra of this Lie algebra.

Formula omitted]-connected components of schemes

A conjecture of Morel asserts that the sheaf π0A1(X) of A1-connected components of a simplicial sheaf X is A1-invariant. A conjecture of Asok and Morel asserts that the sheaves of A1-connected components of smooth schemes over a field coincide with the sheaves of their A1-chain-connected components. Another conjecture of Asok and Morel states that the sheaf of A1-connected components is a birational invariant of smooth proper schemes. In this article, we exhibit examples of schemes for which conjectures of Asok and Morel fail to hold and whose Sing⁎ is not A1-local. We also give equivalent conditions for Morel's conjecture to hold and obtain an explicit conjectural description of π0A1(X). A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field.

Projective modules for the symmetric group and Young's seminormal form

We study the representation theory of the symmetric group Sn in positive characteristic p. Using features of the LLT-algorithm we give a conjectural description of the projective cover P(λ) of the simple module D(λ) where λ is a p-restricted partition such that all ladders of the corresponding ladder partition are of order less than p. Inspired by the recent theory of KLR-algebras, we explain an algorithm that allows us to verify this conjectural description for n≤15, at least.

Yangians and quantizations of slices in the affine Grassmannian

We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for PGL(n) of Finkelberg-Rybnykov.

Kazhdan–Lusztig cells in planar hyperbolic Coxeter groups and automata

Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red (C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red (C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.

Computational Methods for Orbit Closures in a Representation with Finitely Many Orbits

The action of on has finitely many orbits. Kraśkiewicz and Weyman give a conjectural description of the minimal free resolution of the coordinate ring for certain orbit closures in . We verify their conjecture by constructing their complexes explicitly in Macaulay2 and checking that they are exact. We also construct equations for all the orbit closures and verify that they generate radical ideals.

Bound on the Jordan type of a generic nilpotent matrix commuting with a given matrix

It is well known that a nilpotent n×n matrix B is determined up to conjugacy by a partition of n formed by the sizes of the Jordan blocks of B. We call this partition the Jordan type of B. We obtain partial results on the following problem: for any partition P of n describe the type Q(P) of a generic nilpotent matrix commuting with a given nilpotent matrix of type P. A conjectural description for Q(P) was given by P. Oblak and restated by L. Khatami. In this paper we prove “half” of this conjecture by showing that this conjectural type is less than or equal to Q(P) in the dominance order on partitions.

Hypertrees, projections, and moduli of stable rational curves

We give a conjectural description for the cone of effective divisors of the Grothendieck–Knudsen moduli space M̅0,n of stable rational curves with n marked points. Namely, we introduce new combinatorial structures called hypertrees and show that they give exceptional divisors on M̅0,n with many remarkable properties.

A remark on a conjecture of Buzzard-Gee and the cohomology of Shimura varieties

We compare the conjecture of Buzzard-Gee on the association of Galois representations to C-algebraic automorphic representations with the conjectural description of the cohomology of Shimura varieties due to Kottwitz, and the reciprocity law at infinity due to Arthur. This is done by extending Langlands's representation of the L-group associated with a Shimura datum to a representation of the C-group of Buzzard-Gee. The approach offers an explanation of the explicit Tate twist appearing in Kottwitz's description. © International Press 2013.

Tetrahedron and 3D reflection equations from quantized algebra of functions

Soibelman’s theory of quantized function algebra Aq(SLn) provides a representation theoretical scheme to construct a solution of the Zamolodchikov tetrahedron equation. We extend this idea originally due to Kapranov and Voevodsky to Aq(Sp2n) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known three-dimensional (3D) R matrix, the K yields the first ever solution to the 3D analogue of the reflection equation proposed by Isaev and Kulish. It is shown that matrix elements of R and K are polynomials in q and that there are combinatorial and birational counterparts for R and K. The combinatorial ones arise either at q = 0 or by tropicalization of the birational ones. A conjectural description for type B and F4 cases is also given.

Combinatorics of B-orbits and Bruhat–Chevalley order on involutions

Let G be the general linear group, B its standard Borel subgroup, $ \mathfrak{n} $ the space of strictly upper-triangular matrices, $ {\mathfrak{n}^*} $ its dual space, S n the symmetric group on n letters, $ S_n^2 $ the set of involutions of S n . To $ \sigma \in S_n^2 $ one can assign the B-orbit $ {\Omega_\sigma } \subset {\mathfrak{n}^*} $ . Let σ, $ \tau \in S_n^2 $ . We compute dim Ω σ and prove that Ω τ is contained in the Zariski closure of Ω σ if and only if τ ≤ σ with respect to the Bruhat–Chevalley order. We also give a conjectural description of the closure of Ω σ and discuss some connections with the geometry of Schubert varieties.

Boundary Coupling of Lie Algebroid Poisson Sigma Models and Representations up to Homotopy

A general form for the boundary coupling of a Lie algebroid Poisson sigma model is proposed. The approach involves using the Batalin–Vilkovisky formalism in the AKSZ geometrical version, to write a BRST-invariant coupling for a representation up to homotopy of the target Lie algebroid or its subalgebroids. These considerations lead to a conjectural description of topological D-branes on generalized complex manifolds, which includes A-branes and B-branes as special cases.