| 1. |
- Ahlén, Olof, et al.
(author)
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Fourier coefficients attached to small automorphic representations of SLn (A)
- 2018
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In: Journal of Number Theory. - : Elsevier BV. - 0022-314X .- 1096-1658. ; 192, s. 80-142
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Journal article (peer-reviewed)abstract
- We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of SLn(A) are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro–Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.
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| 2. |
- Brubaker, B., et al.
(author)
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Colored five-vertex models and Demazure atoms
- 2021
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In: Journal of Combinatorial Theory Series A. - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 178
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Journal article (peer-reviewed)abstract
- Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a byproduct, we will construct Demazure atoms on Kashiwara's B-infinity crystal and give new algorithms for computing Lascoux-Schutzenberger keys. (C) 2020 Elsevier Inc. All rights reserved.
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| 3. |
- Brubaker, Ben, et al.
(author)
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Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker Functions
- 2020
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In: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 1432-0916 .- 0010-3616. ; 380:2, s. 535-579
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Journal article (peer-reviewed)abstract
- We show that spherical Whittaker functions on an n-fold cover of the general linear group arise naturally from the quantum Fock space representation of Uq(sl^ (n)) introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering the solvable lattice models known as “metaplectic ice” whose partition functions are metaplectic Whittaker functions. First, we show that a certain Hecke action on metaplectic Whittaker coinvariants agrees (up to twisting) with a Hecke action of Ginzburg, Reshetikhin, and Vasserot arising in quantum affine Schur-Weyl duality. This allows us to expand the framework of KMS by Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are necessary for connections to metaplectic forms. Our main theorem interprets the row transfer matrices of this ice model as “half” vertex operators on quantum Fock space that intertwine with the action of Uq(sl^ (n)). In the process, we introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions on an n-fold metaplectic cover of GL r. These resemble LLT polynomials or ribbon symmetric functions introduced by Lascoux, Leclerc and Thibon, and in fact the metaplectic symmetric functions are (up to twisting) specializations of supersymmetric LLT polynomials defined by Lam. Indeed Lam constructed families of symmetric functions from Heisenberg algebra actions on the Fock space commuting with the Uq(sl^ (n)) -action. The Heisenberg algebra is independent of Drinfeld twisting of the quantum group. We explain that half vertex operators agree with Lam’s construction and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models.
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| 4. |
- Fleig, Philipp, et al.
(author)
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Eisenstein series and automorphic representations
- 2015
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Journal article (other academic/artistic)abstract
- We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of Langlands' constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker vector associated to unramified automorphic representations of G(Q_p). Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore introduce also some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics where automorphic forms occur in string theory.
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| 5. |
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| 6. |
- Gourevitch, D., et al.
(author)
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A reduction principle for Fourier coefficients of automorphic forms
- 2022
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In: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 0025-5874 .- 1432-1823. ; 300, s. 2679-2717
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Journal article (peer-reviewed)abstract
- We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic groupG(AK), which we refer to as Fourier coefficients associated to the data of a 'Whittaker pair'. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are 'Levi-distinguished' Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a K-distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.
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| 7. |
- Gourevitch, D., et al.
(author)
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EULERIANITY OF FOURIER COEFFICIENTS OF AUTOMORPHIC FORMS
- 2021
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In: Representation Theory. - : American Mathematical Society (AMS). - 1088-4165. ; 25, s. 481-507
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Journal article (peer-reviewed)abstract
- We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a `hidden' invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.
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| 8. |
- Gourevitch, Dmitry, et al.
(author)
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Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups
- 2022
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In: Canadian Journal of Mathematics. - 1496-4279 .- 0008-414X. ; 74:1, s. 122-169
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Journal article (peer-reviewed)abstract
- In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let ππ be a minimal or next-to-minimal automorphic representation of G. We prove that any η∈πη∈π is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on GLnGLn . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type D5D5 and E8E8 with a view toward applications to scattering amplitudes in string theory.
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| 9. |
- Gustafsson, Henrik, 1988
(author)
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Automorphic forms and string theory: Small automorphic representations and non-perturbative effects
- 2017
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Doctoral thesis (other academic/artistic)abstract
- This compilation thesis stems from a project with the purpose of determining non-perturbative contributions to scattering amplitudes in string theory carrying important information about instantons, black hole quantum states and M-theory. The scattering amplitudes are functions on the moduli space invariant under the discrete U-duality group and this invariance is one of the defining properties of an automorphic form. In particular, the leading terms of the low-energy expansion of four-graviton scattering amplitudes in toroidal compactifications of type IIB string theory are captured by automorphic forms attached to small automorphic representations and their Fourier coefficients describe both perturbative and non-perturbative contributions.In this thesis, Fourier coefficients of automorphic forms attached to small automorphic representations of higher-rank groups are computed with respect to different unipotent subgroups allowing for the study of different types of non-perturbative effects. The analysis makes extensive use of the vanishing properties obtained from supersymmetry described by the global wave-front set of the automorphic representation. Specifically, expressions for Fourier coefficients of automorphic forms attached to a minimal or next-to-minimal automorphic representation of SLn, with respect to the unipotent radicals of maximal parabolic subgroups, are presented in terms of degenerate Whittaker coefficients. Additionally, it is shown how such an automorphic form is completely determined by these Whittaker coefficients.The thesis also includes some partial results for automorphic forms attached to small automorphic representations of E6, E7 and E8.
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| 10. |
- Gustafsson, Henrik, 1988
(author)
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Automorphic string amplitudes
- 2015
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Licentiate thesis (other academic/artistic)abstract
- This thesis explores the non-perturbative properties of higher derivative interactions appearing in the low-energy expansion of four-graviton scattering amplitudes in toroidal compactifications of type IIB string theory. We summarise the arguments for finding such higher derivative corrections in terms of automorphic forms using U-duality, supersymmetry and string perturbation theory. The perturbative and non-perturbative parts can then be studied from their Fourier expansions. To be able to compute such Fourier coefficients we use the adelic framework as an intermediate step which also gives a new perspective on the arithmetic content of the scattering amplitudes.We give a review of known methods for computing certain classes of Fourier coefficients from the mathematical literature as presented in Paper I of the appended papers, and of our own work in Paper II towards computing some of the remaining coefficients of interest in string theory.
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