---
_id: '11545'
abstract:
- lang: eng
text: "We classify contravariant pairings between standard Whittaker modules and
Verma modules over a complex semisimple Lie algebra. These contravariant pairings
are useful in extending several classical techniques for category O to the Miličić–Soergel
category N . We introduce a class of costandard modules which generalize dual
Verma modules, and describe canonical maps from standard to costandard modules
in terms of contravariant pairings.\r\nWe show that costandard modules have unique
irreducible submodules and share the same composition factors as the corresponding
standard Whittaker modules. We show that costandard modules give an algebraic
characterization of the global sections of costandard twisted Harish-Chandra sheaves
on the associated flag variety, which are defined using holonomic duality of D-modules.
We prove that with these costandard modules, blocks of category\r\nN have the
structure of highest weight categories and we establish a BGG reciprocity theorem
for N ."
acknowledgement: We thank Catharina Stroppel and Jens Niklas Eberhardt for interesting
discussions. The first author acknowledges the support of the European Union's Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement
No. 754411. The second author is supported by the National Science Foundation Award
No. 1803059 and the Australian Research Council grant DP170101579.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Anna
full_name: Romanov, Anna
last_name: Romanov
citation:
ama: Brown A, Romanov A. Contravariant pairings between standard Whittaker modules
and Verma modules. Journal of Algebra. 2022;609(11):145-179. doi:10.1016/j.jalgebra.2022.06.017
apa: Brown, A., & Romanov, A. (2022). Contravariant pairings between standard
Whittaker modules and Verma modules. Journal of Algebra. Elsevier. https://doi.org/10.1016/j.jalgebra.2022.06.017
chicago: Brown, Adam, and Anna Romanov. “Contravariant Pairings between Standard
Whittaker Modules and Verma Modules.” Journal of Algebra. Elsevier, 2022.
https://doi.org/10.1016/j.jalgebra.2022.06.017.
ieee: A. Brown and A. Romanov, “Contravariant pairings between standard Whittaker
modules and Verma modules,” Journal of Algebra, vol. 609, no. 11. Elsevier,
pp. 145–179, 2022.
ista: Brown A, Romanov A. 2022. Contravariant pairings between standard Whittaker
modules and Verma modules. Journal of Algebra. 609(11), 145–179.
mla: Brown, Adam, and Anna Romanov. “Contravariant Pairings between Standard Whittaker
Modules and Verma Modules.” Journal of Algebra, vol. 609, no. 11, Elsevier,
2022, pp. 145–79, doi:10.1016/j.jalgebra.2022.06.017.
short: A. Brown, A. Romanov, Journal of Algebra 609 (2022) 145–179.
date_created: 2022-07-08T11:40:07Z
date_published: 2022-11-01T00:00:00Z
date_updated: 2023-08-03T11:56:30Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1016/j.jalgebra.2022.06.017
ec_funded: 1
external_id:
isi:
- '000861841100004'
file:
- access_level: open_access
checksum: 82abaee3d7837f703e499a9ecbb25b7c
content_type: application/pdf
creator: dernst
date_created: 2023-02-02T07:32:48Z
date_updated: 2023-02-02T07:32:48Z
file_id: '12473'
file_name: 2022_JournalAlgebra_Brown.pdf
file_size: 582962
relation: main_file
success: 1
file_date_updated: 2023-02-02T07:32:48Z
has_accepted_license: '1'
intvolume: ' 609'
isi: 1
issue: '11'
keyword:
- Algebra and Number Theory
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: 145-179
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Journal of Algebra
publication_identifier:
issn:
- 0021-8693
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Contravariant pairings between standard Whittaker modules and Verma modules
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 609
year: '2022'
...
---
_id: '7905'
abstract:
- lang: eng
text: We investigate a sheaf-theoretic interpretation of stratification learning
from geometric and topological perspectives. Our main result is the construction
of stratification learning algorithms framed in terms of a sheaf on a partially
ordered set with the Alexandroff topology. We prove that the resulting decomposition
is the unique minimal stratification for which the strata are homogeneous and
the given sheaf is constructible. In particular, when we choose to work with the
local homology sheaf, our algorithm gives an alternative to the local homology
transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM
Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the
cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2),
195–222, 2020). Additionally, we give examples of stratifications based on the
geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018),
illustrating how the sheaf-theoretic approach can be used to study stratifications
from both topological and geometric perspectives. This approach also points toward
future applications of sheaf theory in the study of topological data analysis
by illustrating the utility of the language of sheaf theory in generalizing existing
algorithms.
acknowledgement: Open access funding provided by Institute of Science and Technology
(IST Austria). This work was partially supported by NSF IIS-1513616 and NSF ABI-1661375.
The authors would like to thank the anonymous referees for their insightful comments.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Bei
full_name: Wang, Bei
last_name: Wang
citation:
ama: Brown A, Wang B. Sheaf-theoretic stratification learning from geometric and
topological perspectives. Discrete and Computational Geometry. 2021;65:1166-1198.
doi:10.1007/s00454-020-00206-y
apa: Brown, A., & Wang, B. (2021). Sheaf-theoretic stratification learning from
geometric and topological perspectives. Discrete and Computational Geometry.
Springer Nature. https://doi.org/10.1007/s00454-020-00206-y
chicago: Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from
Geometric and Topological Perspectives.” Discrete and Computational Geometry.
Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00206-y.
ieee: A. Brown and B. Wang, “Sheaf-theoretic stratification learning from geometric
and topological perspectives,” Discrete and Computational Geometry, vol.
65. Springer Nature, pp. 1166–1198, 2021.
ista: Brown A, Wang B. 2021. Sheaf-theoretic stratification learning from geometric
and topological perspectives. Discrete and Computational Geometry. 65, 1166–1198.
mla: Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric
and Topological Perspectives.” Discrete and Computational Geometry, vol.
65, Springer Nature, 2021, pp. 1166–98, doi:10.1007/s00454-020-00206-y.
short: A. Brown, B. Wang, Discrete and Computational Geometry 65 (2021) 1166–1198.
date_created: 2020-05-30T10:26:04Z
date_published: 2021-06-01T00:00:00Z
date_updated: 2024-03-07T15:01:58Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-020-00206-y
external_id:
arxiv:
- '1712.07734'
isi:
- '000536324700001'
file:
- access_level: open_access
checksum: 487a84ea5841b75f04f66d7ebd71b67e
content_type: application/pdf
creator: dernst
date_created: 2020-11-25T09:06:41Z
date_updated: 2020-11-25T09:06:41Z
file_id: '8803'
file_name: 2020_DiscreteCompGeometry_Brown.pdf
file_size: 1013730
relation: main_file
success: 1
file_date_updated: 2020-11-25T09:06:41Z
has_accepted_license: '1'
intvolume: ' 65'
isi: 1
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 1166-1198
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sheaf-theoretic stratification learning from geometric and topological perspectives
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 65
year: '2021'
...
---
_id: '8773'
abstract:
- lang: eng
text: Let g be a complex semisimple Lie algebra. We give a classification of contravariant
forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We
prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose
dimension is given by the cardinality of the Weyl group of g. We also describe
a procedure for parabolically inducing contravariant forms. As a corollary, we
deduce the existence of the Shapovalov form on a Verma module, and provide a formula
for the dimension of the space of contravariant forms on the degenerate Whittaker
modules M(χ,η) introduced by McDowell.
acknowledgement: "We would like to thank Peter Trapa for useful discussions, and Dragan
Milicic and Arun Ram for valuable feedback on the structure of the paper. The first
author acknowledges the support of the European Unions Horizon 2020 research and
innovation programme under the Marie Skodowska-Curie Grant Agreement No. 754411.
The second author is\r\nsupported by the National Science Foundation Award No. 1803059."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Anna
full_name: Romanov, Anna
last_name: Romanov
citation:
ama: Brown A, Romanov A. Contravariant forms on Whittaker modules. Proceedings
of the American Mathematical Society. 2021;149(1):37-52. doi:10.1090/proc/15205
apa: Brown, A., & Romanov, A. (2021). Contravariant forms on Whittaker modules.
Proceedings of the American Mathematical Society. American Mathematical
Society. https://doi.org/10.1090/proc/15205
chicago: Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.”
Proceedings of the American Mathematical Society. American Mathematical
Society, 2021. https://doi.org/10.1090/proc/15205.
ieee: A. Brown and A. Romanov, “Contravariant forms on Whittaker modules,” Proceedings
of the American Mathematical Society, vol. 149, no. 1. American Mathematical
Society, pp. 37–52, 2021.
ista: Brown A, Romanov A. 2021. Contravariant forms on Whittaker modules. Proceedings
of the American Mathematical Society. 149(1), 37–52.
mla: Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.”
Proceedings of the American Mathematical Society, vol. 149, no. 1, American
Mathematical Society, 2021, pp. 37–52, doi:10.1090/proc/15205.
short: A. Brown, A. Romanov, Proceedings of the American Mathematical Society 149
(2021) 37–52.
date_created: 2020-11-19T10:17:40Z
date_published: 2021-01-01T00:00:00Z
date_updated: 2023-08-04T11:11:47Z
day: '01'
department:
- _id: HeEd
doi: 10.1090/proc/15205
ec_funded: 1
external_id:
arxiv:
- '1910.08286'
isi:
- '000600416300004'
intvolume: ' 149'
isi: 1
issue: '1'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1910.08286
month: '01'
oa: 1
oa_version: Preprint
page: 37-52
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Proceedings of the American Mathematical Society
publication_identifier:
eissn:
- 1088-6826
issn:
- 0002-9939
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: Contravariant forms on Whittaker modules
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 149
year: '2021'
...
---
_id: '9111'
abstract:
- lang: eng
text: 'We study the probabilistic convergence between the mapper graph and the Reeb
graph of a topological space X equipped with a continuous function f:X→R. We first
give a categorification of the mapper graph and the Reeb graph by interpreting
them in terms of cosheaves and stratified covers of the real line R. We then introduce
a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium
on point-based graphics, 2007), referred to as the enhanced mapper graph, and
demonstrate that such a construction approximates the Reeb graph of (X,f) when
it is applied to points randomly sampled from a probability density function concentrated
on (X,f). Our techniques are based on the interleaving distance of constructible
cosheaves and topological estimation via kernel density estimates. Following Munch
and Wang (In: 32nd international symposium on computational geometry, volume 51
of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany,
pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible
R-space (with a fixed open cover), approximates the Reeb graph of the same space.
We then construct an isomorphism between the mapper of (X,f) to the mapper of
a super-level set of a probability density function concentrated on (X,f). Finally,
building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b),
we show that, with high probability, we can recover the mapper of the super-level
set given a sufficiently large sample. Our work is the first to consider the mapper
construction using the theory of cosheaves in a probabilistic setting. It is part
of an ongoing effort to combine sheaf theory, probability, and statistics, to
support topological data analysis with random data.'
acknowledgement: "AB was supported in part by the European Union’s Horizon 2020 research
and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No.
754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation,
Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was
supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like
to thank the Institute for Mathematics and its Applications for hosting a workshop
titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen
Access funding provided by Institute of Science and Technology (IST Austria)."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Omer
full_name: Bobrowski, Omer
last_name: Bobrowski
- first_name: Elizabeth
full_name: Munch, Elizabeth
last_name: Munch
- first_name: Bei
full_name: Wang, Bei
last_name: Wang
citation:
ama: Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability
of random mapper graphs. Journal of Applied and Computational Topology.
2021;5(1):99-140. doi:10.1007/s41468-020-00063-x
apa: Brown, A., Bobrowski, O., Munch, E., & Wang, B. (2021). Probabilistic convergence
and stability of random mapper graphs. Journal of Applied and Computational
Topology. Springer Nature. https://doi.org/10.1007/s41468-020-00063-x
chicago: Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic
Convergence and Stability of Random Mapper Graphs.” Journal of Applied and
Computational Topology. Springer Nature, 2021. https://doi.org/10.1007/s41468-020-00063-x.
ieee: A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence
and stability of random mapper graphs,” Journal of Applied and Computational
Topology, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.
ista: Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and
stability of random mapper graphs. Journal of Applied and Computational Topology.
5(1), 99–140.
mla: Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper
Graphs.” Journal of Applied and Computational Topology, vol. 5, no. 1,
Springer Nature, 2021, pp. 99–140, doi:10.1007/s41468-020-00063-x.
short: A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational
Topology 5 (2021) 99–140.
date_created: 2021-02-11T14:41:02Z
date_published: 2021-03-01T00:00:00Z
date_updated: 2023-09-05T15:37:56Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s41468-020-00063-x
ec_funded: 1
external_id:
arxiv:
- '1909.03488'
file:
- access_level: open_access
checksum: 3f02e9d47c428484733da0f588a3c069
content_type: application/pdf
creator: dernst
date_created: 2021-02-11T14:43:59Z
date_updated: 2021-02-11T14:43:59Z
file_id: '9112'
file_name: 2020_JourApplCompTopology_Brown.pdf
file_size: 2090265
relation: main_file
success: 1
file_date_updated: 2021-02-11T14:43:59Z
has_accepted_license: '1'
intvolume: ' 5'
issue: '1'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 99-140
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Probabilistic convergence and stability of random mapper graphs
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 5
year: '2021'
...
---
_id: '6828'
abstract:
- lang: eng
text: In this paper we construct a family of exact functors from the category of
Whittaker modules of the simple complex Lie algebra of type to the category of
finite-dimensional modules of the graded affine Hecke algebra of type . Using
results of Backelin [2] and of Arakawa-Suzuki [1], we prove that these functors
map standard modules to standard modules (or zero) and simple modules to simple
modules (or zero). Moreover, we show that each simple module of the graded affine
Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker
category contains the BGG category as a full subcategory, our results generalize
results of Arakawa-Suzuki [1], which in turn generalize Schur-Weyl duality between
finite-dimensional representations of and representations of the symmetric group
.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
citation:
ama: Brown A. Arakawa-Suzuki functors for Whittaker modules. Journal of Algebra.
2019;538:261-289. doi:10.1016/j.jalgebra.2019.07.027
apa: Brown, A. (2019). Arakawa-Suzuki functors for Whittaker modules. Journal
of Algebra. Elsevier. https://doi.org/10.1016/j.jalgebra.2019.07.027
chicago: Brown, Adam. “Arakawa-Suzuki Functors for Whittaker Modules.” Journal
of Algebra. Elsevier, 2019. https://doi.org/10.1016/j.jalgebra.2019.07.027.
ieee: A. Brown, “Arakawa-Suzuki functors for Whittaker modules,” Journal of Algebra,
vol. 538. Elsevier, pp. 261–289, 2019.
ista: Brown A. 2019. Arakawa-Suzuki functors for Whittaker modules. Journal of Algebra.
538, 261–289.
mla: Brown, Adam. “Arakawa-Suzuki Functors for Whittaker Modules.” Journal of
Algebra, vol. 538, Elsevier, 2019, pp. 261–89, doi:10.1016/j.jalgebra.2019.07.027.
short: A. Brown, Journal of Algebra 538 (2019) 261–289.
date_created: 2019-08-22T07:54:13Z
date_published: 2019-11-15T00:00:00Z
date_updated: 2023-08-29T07:11:47Z
day: '15'
department:
- _id: HeEd
doi: 10.1016/j.jalgebra.2019.07.027
external_id:
arxiv:
- '1805.04676'
isi:
- '000487176300011'
intvolume: ' 538'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1805.04676
month: '11'
oa: 1
oa_version: Preprint
page: 261-289
publication: Journal of Algebra
publication_identifier:
issn:
- 0021-8693
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: Arakawa-Suzuki functors for Whittaker modules
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 538
year: '2019'
...