Theta Operator Research Articles

Delta and Theta Operator Expansions

Abstract We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking functions. Here, $\Delta _F$ and $\Pi $ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$ . Our main results, in turn, give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta _{Fe_1} \Theta _{G} J$ whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis $\{\Pi e_\lambda ^\ast \}_\lambda $ . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.

New identities for theta operators

In this article, we prove a new general identity involving the Theta operators introduced by the first author, Iraci, and Vanden Wyngaerd [Adv. Math. 376 (2021), p.59]. From this result, we can easily deduce several new identities that have combinatorial consequences in the study of Macdonald polynomials and diagonal coinvariants. In particular, we provide a unifying framework from which we recover many identities scattered in the literature, often resulting in drastically shorter proofs.

A proof of the fermionic theta coinvariant conjecture

Let (x1,…,xn,y1,…,yn) be a list of 2n commuting variables, (θ1,…,θn,ξ1,…,ξn) be a list of 2n anticommuting variables, and C[xn,yn]⊗∧{θn,ξn} be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded Sn-isomorphism type of C[xn,yn]⊗∧{θn,ξn}/I where I is the ideal generated by Sn-invariants with vanishing constant term. We prove their conjecture in the ‘purely fermionic setting’ obtained by setting the commuting variables xi,yi equal to zero.

The theta cycles for modular forms modulo prime powers

Abstract Recently, Chen and Kiming studied the theta operator on modular forms modulo prime powers p m {p^{m}} , where p ≥ 5 {p\geq 5} and m ≥ 2 {m\geq 2} . In this paper, we study mod p m {p^{m}} filtrations and mod p m {p^{m}} theta cycles. We give a bound on some elements in the mod p m {p^{m}} theta cycle ( m ≥ 2 {m\geq 2} ), and we exactly compute those values in the case that m = 2 {m=2} .

Tiered Trees and Theta Operators

Abstract In [10], the authors introduced tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers and torus orbits on certain homogeneous varieties. In this paper, we use Theta operators, introduced in [6], to give a symmetric function formula that enumerates these trees. We then formulate a general conjecture that extends this result, a special case of which might give some insight about how to formulate a unified Delta conjecture [20].

Reductions of Galois representations and the theta operator

Let [Formula: see text] be a prime, and let [Formula: see text] be a cuspidal eigenform of weight at least [Formula: see text] and level coprime to [Formula: see text] of finite slope [Formula: see text]. Let [Formula: see text] denote the mod [Formula: see text] Galois representation associated with [Formula: see text] and [Formula: see text] the mod [Formula: see text] cyclotomic character. Under an assumption on the weight of [Formula: see text], we prove that there exists a cuspidal eigenform [Formula: see text] of weight at least [Formula: see text] and level coprime to [Formula: see text] of slope [Formula: see text] such that [Formula: see text] up to semisimplification. The proof uses Hida–Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions of the local Galois representations associated to cusp forms with slopes in the interval [Formula: see text] were determined by Deligne, Buzzard and Gee and for slopes in [Formula: see text] by Bhattacharya, Ganguli, Ghate, Rai and Rozensztajn. We show that these reductions, in spite of their somewhat complicated behavior, are compatible with the displayed equation above. Moreover, the displayed equation above allows us to predict the shape of the reductions of a class of Galois representations attached to eigenforms of slope larger than [Formula: see text]. Finally, the methods of this paper allow us to obtain upper bounds on the radii of certain Coleman families.

A proof of the compositional Delta conjecture

We prove a compositional refinement of the Delta conjecture (rise version) of Haglund, Remmel and Wilson [16] for Δen−k−1′en which was stated in [8] in terms of Theta operators.

Differential operators mod p : analytic continuation and consequences

This paper concerns certain $\mod p$ differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the $\mod p$ reduction of the $p$-adic theta operators previously studied by some of the authors. In the characteristic $0$, $p$-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole Shimura variety. On the other hand, our $\mod p$ operators extend ("analytically continue", in the language of de Shalit and Goren) to the whole Shimura variety. As a consequence, motivated by their use by Edixhoven and Jochnowitz in the case of modular forms for proving the weight part of Serre's conjecture, we discuss some effects of these operators on Galois representations. Our focus and techniques differ from those in the literature. Our intrinsic, coordinate-free approach removes difficulties that arise from working with $q$-expansions and works in settings where earlier techniques, which rely on explicit calculations, are not applicable. In contrast with previous constructions and analytic continuation results, these techniques work for any totally real base field, any weight, and all signatures and ranks of groups at once, recovering prior results on analytic continuation as special cases.

A proof of the theta operator conjecture

In the context of the (generalized) Delta Conjecture and its compositional form, D'Adderio, Iraci, and Vanden Wyngaerd recently stated a conjecture relating two symmetric function operators, Dk and Θk. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the Dk operators. Even though we deal specifically with the relation between the Dk and Θk operators, our work introduces a method for finding relations between Θk and other plethystic operators which are important in this area of study.

Entire Theta Operators at Unramified Primes

Abstract Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\textrm{mod}\; p$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes $p$ to the whole Shimura variety of the $\textrm{mod}\; p$ reduction of $p$-adic Maass–Shimura operators a priori defined only over the $\mu $-ordinary locus, and (2) the construction of new $\textrm{mod}\; p$ theta operators that do not arise as the $\textrm{mod}\; p$ reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree.

A Valley Version of the Delta Square Conjecture

Inspired by [Qiu, Wilson 2019] and [D'Adderio, Iraci, Vanden Wyngaerd 2019 - Delta Square], we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in [Haglund, Sergel 2019] to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in [Sergel 2016], relying on the formulation of the touching version in terms of the $\Theta_f$ operators introduced in [D'Adderio, Iraci, Vanden Wyngaerd 2019 - Theta Operators].

Theta operators, refined Delta conjectures, and coinvariants

We introduce the family of Theta operators Θf indexed by symmetric functions f that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson [23] for Δen−k−1′en. We show that the 4-variable Catalan theorem of Zabrocki [31] is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit in [6], extending one of their results.Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for ΔhmΔen−k−1′en, and prove the case k=0, which was also conjectured in [23], extending the shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for Δhm∇en. Moreover we show how this implies the case k=0 of our generalized Delta square conjecture for [n−k]t[n]tΔhmΔen−kω(pn), extending the square theorem of Sergel [27] to a generalized square theorem for Δhm∇ω(pn).Still the Theta operators will provide a conjectural formula for the Frobenius characteristic of super-diagonal coinvariants with two sets of Grassmannian variables, extending the one of Zabrocki in [30] for the case with one set of such variables. We propose a combinatorial interpretation of this last formula at q=1, leaving open the problem of finding a dinv statistic that gives the whole symmetric function.

Theta operators on unitary Shimura varieties

We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the {\mu}-ordinary locus, when applied to scalar-valued forms.

Theta operator on Hermitian modular forms over the Eisenstein field

The mod p kernel of the theta operator on Hermitian modular forms is studied in the case that the base field is the Eisenstein field.

Weights of the Mod p Kernels of Theta Operators

AbstractLet Θ[j] be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime p, we give the weights of elements of mod p kernel of Θ[j], where the mod p kernel of Θ[j] is the set of all Siegel modular forms F such that Θ[j](F) is congruent to zero modulo p. In order to construct examples of the mod p kernel of Θ[j] fromany Siegel modular form, we introduce new operators A(j)(M) and show the modularity of F|A(j)(M) when F is a Siegel modular form. Finally, we give some examples of the mod p kernel of Θ[j] and the filtrations of some of them.

On the mod p kernel of the theta operator and Eisenstein series

Siegel modular forms in the space of the mod p kernel of the theta operator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given.

On the kernel of the theta operator mod p

We construct many examples of level one Siegel modular forms in the kernel of theta operators mod p by using theta series attached to positive definite quadratic forms.

Theta operators, Goss polynomials, and v-adic modular forms

We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality.

On the theta operator for Hermitian modular forms of degree 2

The mod p kernel of the theta operator is the set of modular forms whose image of the theta operator is congruent to zero modulo a prime p. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa’s odd weight cusp form is an element of mod 23 kernel of the theta operator. In this paper, we give some examples which represent elements in the mod p kernel of the theta operator in the case of Hermitian modular forms of degree 2.

A theta operator on Picard modular forms modulo an inert prime

To thememory of Robert Coleman *Correspondence: deshalit@math.huji.ac.il 1Hebrew University, Jerusalem, Israel Full list of author information is available at the end of the article