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Rationality of twist representation zeta functions of compact p-adic analytic groups (2024)
Journal Article
Stasinski, A., & Zordan, M. (2024). Rationality of twist representation zeta functions of compact p-adic analytic groups. Transactions of the American Mathematical Society, 377, 7601-7631

We prove that for any twist rigid compact p-adic analytic group G, its twist representation zeta function is a finite sum of terms n −s i f i (p −s), where n i are natural numbers and f i (t) ∈ Q(t) are rational functions. Mero-morphic continuation a... Read More about Rationality of twist representation zeta functions of compact p-adic analytic groups.

Rationality of representation zeta functions of compact p-adic analytic groups (2023)
Journal Article
Stasinski, A., & Zordan, M. (in press). Rationality of representation zeta functions of compact p-adic analytic groups. American Journal of Mathematics,

We prove that for any FAb compact p-adic analytic group G, its representation zeta function is a finite sum of terms n −s i fi(p −s), where ni are natural numbers and fi(t) ∈ Q(t) are rational functions. Meromorphic continuation and rationality of th... Read More about Rationality of representation zeta functions of compact p-adic analytic groups.

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes (2023)
Journal Article
Knight, L., & Stasinski, A. (2023). Representatives of similarity classes of matrices over PIDs corresponding to ideal classes. Glasgow Mathematical Journal, 66(1), 88-103. https://doi.org/10.1017/s0017089523000356

For a principal ideal domain A, the Latimer–MacDuffee correspondence sets up a bijection between the similarity classes of matrices in Mn(A) with irreducible characteristic polynomial f(x) and the ideal classes of the order A[x]/(f(x)). We prove that... Read More about Representatives of similarity classes of matrices over PIDs corresponding to ideal classes.

A uniform proof of the finiteness of the class group of a global field (2021)
Journal Article
Stasinski, A. (2021). A uniform proof of the finiteness of the class group of a global field. The American Mathematical Monthly, 128(3), 239-249. https://doi.org/10.1080/00029890.2021.1855036

We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of rings of... Read More about A uniform proof of the finiteness of the class group of a global field.

Representations of SL over finite local rings of length two (2020)
Journal Article
Stasinski, A. (2021). Representations of SL over finite local rings of length two. Journal of Algebra, 566, 119-135. https://doi.org/10.1016/j.jalgebra.2020.08.036

Let Fqbe a finite field of characteristic pand let W2(Fq)be the ring of Witt vectors of length two over Fq. We prove that for any integer nsuch that pdivides n, the groups SLn(Fq[t]/t2)and SLn(W2(Fq)) have the same number of irreducible representatio... Read More about Representations of SL over finite local rings of length two.

Representation growth of compact linear groups (2019)
Journal Article
Häsä, J., & Stasinski, A. (2019). Representation growth of compact linear groups. Transactions of the American Mathematical Society, 372(2), 925-980. https://doi.org/10.1090/tran/7618

We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GLn(O). This amounts to a study of the abscissae of convergence of the correspo... Read More about Representation growth of compact linear groups.

Representations of reductive groups over finite local rings of length two (2018)
Journal Article
Stasinski, A., & Vera-Gajardo, A. (2019). Representations of reductive groups over finite local rings of length two. Journal of Algebra, 525, 171-190. https://doi.org/10.1016/j.jalgebra.2018.11.039

LetFqbe a finite field of characteristicp, and letW2(Fq)be thering of Witt vectors of length two overFq. We prove that for any reduc-tive group schemeGoverZsuch thatpis very good forG×Fq, the groupsG(Fq[t]/t2)andG(W2(Fq))have the same number of irred... Read More about Representations of reductive groups over finite local rings of length two.

Commutators of trace zero matrices over principal ideal rings (2018)
Journal Article
Stasinski, A. (2018). Commutators of trace zero matrices over principal ideal rings. Israel Journal of Mathematics, 228(1), 211-227. https://doi.org/10.1007/s11856-018-1762-5

We prove that for every trace zero square matrix A of size at least 3 over a principal ideal ring R, there exist trace zero matrices X, Y over R such that XY−YX = A. Moreover, we show that X can be taken to be regular mod every maximal ideal of R. Th... Read More about Commutators of trace zero matrices over principal ideal rings.

The regular representations of GLN over finite local principal ideal rings (2017)
Journal Article
Stasinski, A., & Stevens, S. (2017). The regular representations of GLN over finite local principal ideal rings. Bulletin of the London Mathematical Society, 49(6), 1066-1084. https://doi.org/10.1112/blms.12099

Let o o be the ring of integers in a non-Archimedean local field with finite residue field, p p its maximal ideal, and r ⩾ 2 r⩾2 an integer. An irreducible representation of the finite group G r = GL N ( o / p r ) Gr=GLN(o/pr), for an integer N ⩾ 2 N... Read More about The regular representations of GLN over finite local principal ideal rings.

The algebraisation of higher Deligne–Lusztig representations (2017)
Journal Article
Chen, Z., & Stasinski, A. (2017). The algebraisation of higher Deligne–Lusztig representations. Selecta Mathematica (New Series), 23(4), 2907-2926. https://doi.org/10.1007/s00029-017-0349-z

In this paper we study higher Deligne–Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations, defined by Lusztig, coincide with certa... Read More about The algebraisation of higher Deligne–Lusztig representations.

Representations of GL_N over finite local principal ideal rings: an overview (2017)
Presentation / Conference Contribution
Stasinski, A. (2015, June). Representations of GL_N over finite local principal ideal rings: an overview. Presented at Around Langlands Correspondences, Paris, France

We give a survey of the representation theory of GLN over finite local principal ideal rings via Clifford theory, with an emphasis on the construction of regular representations. We review results of Shintani and Hill, and the generalisation of Takas... Read More about Representations of GL_N over finite local principal ideal rings: an overview.

Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces (2016)
Journal Article
Stasinski, A., & Voll, C. (2017). Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces. Forum Mathematicum, 29(3), 717-734. https://doi.org/10.1515/forum-2015-0099

We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled o... Read More about Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces.

Similarity and commutators of matrices over principal ideal rings (2015)
Journal Article
Stasinski, A. (2016). Similarity and commutators of matrices over principal ideal rings. Transactions of the American Mathematical Society, 368(4), 2333-2354. https://doi.org/10.1090/tran/6402

We prove that if is a principal ideal ring and is a matrix with trace zero, then is a commutator, that is, for some . This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over due to Laffey and Reams, a... Read More about Similarity and commutators of matrices over principal ideal rings.

Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B (2014)
Journal Article
Stasinski, A., & Voll, C. (2014). Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B. American Journal of Mathematics, 136(2), 501-550. https://doi.org/10.1353/ajm.2014.0010

We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and $\frak{p}$-adic in... Read More about Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B.

A New Statistic on the Hyperoctahedral Groups (2013)
Journal Article
Stasinski, A., & Voll, C. (2013). A New Statistic on the Hyperoctahedral Groups. Electronic Journal of Combinatorics, 20(3), Article 50

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, an... Read More about A New Statistic on the Hyperoctahedral Groups.

Reductive group schemes, the Greenberg functor, and associated algebraic groups (2012)
Journal Article
Stasinski, A. (2012). Reductive group schemes, the Greenberg functor, and associated algebraic groups. Journal of Pure and Applied Algebra, 216(5), 1092-1101. https://doi.org/10.1016/j.jpaa.2011.10.027

Let AA be an Artinian local ring with algebraically closed residue field kk, and let View the MathML sourceG be an affine smooth group scheme over AA. The Greenberg functor FF associates to View the MathML sourceG a linear algebraic group View the Ma... Read More about Reductive group schemes, the Greenberg functor, and associated algebraic groups.

Extended Deligne–Lusztig varieties for general and special linear groups (2011)
Journal Article
Stasinski, A. (2011). Extended Deligne–Lusztig varieties for general and special linear groups. Advances in Mathematics, 226(3), 2825-2853. https://doi.org/10.1016/j.aim.2010.10.010

We give a generalisation of Deligne–Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieti... Read More about Extended Deligne–Lusztig varieties for general and special linear groups.

On cuspidal Representations of General Linear Groups over Discrete Valuation Rings (2010)
Journal Article
Aubert, A., Onn, U., Prasad, A., & Stasinski, A. (2010). On cuspidal Representations of General Linear Groups over Discrete Valuation Rings. Israel Journal of Mathematics, 175(1), 391-420. https://doi.org/10.1007/s11856-010-0016-y

We define a new notion of cuspidality for representations of GL n over a finite quotient o k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G λ of t... Read More about On cuspidal Representations of General Linear Groups over Discrete Valuation Rings.

Unramified representations of reductive groups over finite rings (2009)
Journal Article
Stasinski, A. (2009). Unramified representations of reductive groups over finite rings. Representation Theory, 13, 636-656. https://doi.org/10.1090/s1088-4165-09-00350-1

Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic $ p$, extending the construction of Deligne and Lusztig of representations of reductive groups over finite field... Read More about Unramified representations of reductive groups over finite rings.