Carlos André - Academia.edu (original) (raw)

Papers by Carlos André

Research paper thumbnail of Supercharacter theories for finite linear p-groups

The group of unipotent uppertriangular matrices with coefficients in a finite field is famous for... more The group of unipotent uppertriangular matrices with coefficients in a finite field is famous for having a completely intractable character theory. Neither the conjugacy classes, nor the irreducible characters, can be given in any generality. By taking certain unions of conjugacy classes, and sums of irreducible characters, a tractable theory is achieved. The resulting ”superclasses” and ”supercharacter” are indexed by set partitions and have combinatorics similar to the Young tableaux of the symmetric group. A similar theory is developed for maximal unipotent subgroups of other classical groups (joint work with Ana M. Neto).

Research paper thumbnail of Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

Journal of Group Theory, 2021

We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-d... more We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

Research paper thumbnail of Basic Characters of the Unitriangular Group

Research paper thumbnail of Super-characters of finite unipotent groups of types Bn, Cn and Dn

Journal of Algebra, 2006

We define and study super-characters (over the complex field) of the classical finite unipotent g... more We define and study super-characters (over the complex field) of the classical finite unipotent groups of types B n , C n and D n. Under the assumption that the prime is sufficiently large, we extend the known results for the unitriangular group proved by the first author in the papers: [C.A.M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995) 287-319], and [C.A.M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130 (7) (2002) 1943-1954]. In particular, we prove that every irreducible (complex) character occurs as a constituent of a unique super-character. We also give a combinatorial description of all the irreducible characters of maximum degree.

Research paper thumbnail of On the Coadjoint Orbits of the Unitriangular Group

Research paper thumbnail of Irreducible characters of groups associated with finite nilpotent algebras with involution

Journal of Algebra, 2010

An algebra group is a group of the form P = 1 + J where J is a finite-dimensional nilpotent assoc... more An algebra group is a group of the form P = 1 + J where J is a finite-dimensional nilpotent associative algebra. A theorem of Z. Halasi asserts that, in the case where J is defined over a finite field F , every irreducible character of P is induced from a linear character of an algebra subgroup of P. If (J , σ) is a nilpotent algebra with involution, then σ naturally defines a group automorphism of P = 1 + J , and we may consider the fixed point subgroup C P (σ). Assuming that F has odd characteristic p, we show that every irreducible character of C P (σ) is induced from a linear character of a subgroup of the form C Q (σ) where Q is a σ-invariant algebra subgroup of P. As a particular case, the result holds for the Sylow p-subgroups of the finite classical groups of Lie type.

Research paper thumbnail of Irreducible characters of finite algebra groups

Arxiv preprint math/9811132, 1998

Research paper thumbnail of The Regular Character of the Unitriangular Group

Research paper thumbnail of Supercharacters of the sylow p-subgroups of the finite symplectic and orthogonal groups

Pacific Journal of Mathematics, 2009

We define and study supercharacters of the classical finite unipotent groups of types B n (q), C ... more We define and study supercharacters of the classical finite unipotent groups of types B n (q), C n (q) and D n (q). We show that the results we proved in 2006 remain valid over any finite field of odd characteristic. In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values. As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.

Research paper thumbnail of Supercharacters of discrete algebra groups

The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs in... more The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs in [15] as an alternative to the usual irreducible character theory, and exemplified with a particular construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. Our construction can be understood as a cruder version of Kirillov’s orbit method and a generalisation of Diaconis and Isaacs construction for finite algebra groups. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with the characterisation of the standard supercharacters of the group of upper unitriangular matrices over an algebraic closed field of prime characteristic.

Research paper thumbnail of Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

Advances in Mathematics, 2012

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier an... more We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

Research paper thumbnail of A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

Research paper thumbnail of Basic characters of the unitriangular group (for arbitrary primes)

PROCEEDINGS-AMERICAN MATHEMATICAL …, 2002

Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elem... more Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption p ≥ n, where p is the characteristic of the field Fq.

Research paper thumbnail of Hecke algebras for the basic characters of the unitriangular group

Proceedings of the American Mathematical Society, 2004

Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a ... more Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a previous paper we obtained a decomposition of the regular character of Un(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ξ (ϕ) of Un(q). We prove that ξ (ϕ) is induced from a linear character of an algebra subgroup of Un(q), and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of ξ (ϕ) as characters induced from an algebra subgroup of Un(q). Finally, we identify a special irreducible constituent of ξ (ϕ), which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption p ≥ n where p is the characteristic of the field) that gives a necessary and sufficient condition for ξ (ϕ) to have a unique irreducible constituent.

Research paper thumbnail of A supercharacter theory for involutive algebra groups

Journal of Algebra, 2015

ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, ... more ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, the algebra group P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If mathscrJ\mathscr{J}mathscrJ is endowed with an involution widehatvarsigma\widehat{\varsigma}widehatvarsigma, then widehatvarsigma\widehat{\varsigma}widehatvarsigma naturally defines a group automorphism of P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ, and we may consider the fixed point subgroup CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1. Assuming that Bbbk\BbbkBbbk has odd characteristic ppp, we use the standard supercharacter theory for PPP to construct a supercharacter theory for CP(widehatvarsigma)C_{P}(\widehat{\varsigma})CP(widehatvarsigma). In particular, we obtain a supercharacter theory for the Sylow ppp-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr\'e and Neto for the special case of the symplectic and orthogonal groups.

Research paper thumbnail of A supercharacter theory for involutive algebra groups

Journal of Algebra, 2015

ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, ... more ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, the algebra group P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If mathscrJ\mathscr{J}mathscrJ is endowed with an involution widehatvarsigma\widehat{\varsigma}widehatvarsigma, then widehatvarsigma\widehat{\varsigma}widehatvarsigma naturally defines a group automorphism of P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ, and we may consider the fixed point subgroup CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1. Assuming that Bbbk\BbbkBbbk has odd characteristic ppp, we use the standard supercharacter theory for PPP to construct a supercharacter theory for CP(widehatvarsigma)C_{P}(\widehat{\varsigma})CP(widehatvarsigma). In particular, we obtain a supercharacter theory for the Sylow ppp-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr\'e and Neto for the special case of the symplectic and orthogonal groups.

Research paper thumbnail of Supercharacter theories for finite linear p-groups

The group of unipotent uppertriangular matrices with coefficients in a finite field is famous for... more The group of unipotent uppertriangular matrices with coefficients in a finite field is famous for having a completely intractable character theory. Neither the conjugacy classes, nor the irreducible characters, can be given in any generality. By taking certain unions of conjugacy classes, and sums of irreducible characters, a tractable theory is achieved. The resulting ”superclasses” and ”supercharacter” are indexed by set partitions and have combinatorics similar to the Young tableaux of the symmetric group. A similar theory is developed for maximal unipotent subgroups of other classical groups (joint work with Ana M. Neto).

Research paper thumbnail of Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

Journal of Group Theory, 2021

We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-d... more We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

Research paper thumbnail of Basic Characters of the Unitriangular Group

Research paper thumbnail of Super-characters of finite unipotent groups of types Bn, Cn and Dn

Journal of Algebra, 2006

We define and study super-characters (over the complex field) of the classical finite unipotent g... more We define and study super-characters (over the complex field) of the classical finite unipotent groups of types B n , C n and D n. Under the assumption that the prime is sufficiently large, we extend the known results for the unitriangular group proved by the first author in the papers: [C.A.M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995) 287-319], and [C.A.M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130 (7) (2002) 1943-1954]. In particular, we prove that every irreducible (complex) character occurs as a constituent of a unique super-character. We also give a combinatorial description of all the irreducible characters of maximum degree.

Research paper thumbnail of On the Coadjoint Orbits of the Unitriangular Group

Research paper thumbnail of Irreducible characters of groups associated with finite nilpotent algebras with involution

Journal of Algebra, 2010

An algebra group is a group of the form P = 1 + J where J is a finite-dimensional nilpotent assoc... more An algebra group is a group of the form P = 1 + J where J is a finite-dimensional nilpotent associative algebra. A theorem of Z. Halasi asserts that, in the case where J is defined over a finite field F , every irreducible character of P is induced from a linear character of an algebra subgroup of P. If (J , σ) is a nilpotent algebra with involution, then σ naturally defines a group automorphism of P = 1 + J , and we may consider the fixed point subgroup C P (σ). Assuming that F has odd characteristic p, we show that every irreducible character of C P (σ) is induced from a linear character of a subgroup of the form C Q (σ) where Q is a σ-invariant algebra subgroup of P. As a particular case, the result holds for the Sylow p-subgroups of the finite classical groups of Lie type.

Research paper thumbnail of Irreducible characters of finite algebra groups

Arxiv preprint math/9811132, 1998

Research paper thumbnail of The Regular Character of the Unitriangular Group

Research paper thumbnail of Supercharacters of the sylow p-subgroups of the finite symplectic and orthogonal groups

Pacific Journal of Mathematics, 2009

We define and study supercharacters of the classical finite unipotent groups of types B n (q), C ... more We define and study supercharacters of the classical finite unipotent groups of types B n (q), C n (q) and D n (q). We show that the results we proved in 2006 remain valid over any finite field of odd characteristic. In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values. As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.

Research paper thumbnail of Supercharacters of discrete algebra groups

The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs in... more The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs in [15] as an alternative to the usual irreducible character theory, and exemplified with a particular construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. Our construction can be understood as a cruder version of Kirillov’s orbit method and a generalisation of Diaconis and Isaacs construction for finite algebra groups. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with the characterisation of the standard supercharacters of the group of upper unitriangular matrices over an algebraic closed field of prime characteristic.

Research paper thumbnail of Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

Advances in Mathematics, 2012

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier an... more We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

Research paper thumbnail of A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

Research paper thumbnail of Basic characters of the unitriangular group (for arbitrary primes)

PROCEEDINGS-AMERICAN MATHEMATICAL …, 2002

Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elem... more Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption p ≥ n, where p is the characteristic of the field Fq.

Research paper thumbnail of Hecke algebras for the basic characters of the unitriangular group

Proceedings of the American Mathematical Society, 2004

Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a ... more Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a previous paper we obtained a decomposition of the regular character of Un(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ξ (ϕ) of Un(q). We prove that ξ (ϕ) is induced from a linear character of an algebra subgroup of Un(q), and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of ξ (ϕ) as characters induced from an algebra subgroup of Un(q). Finally, we identify a special irreducible constituent of ξ (ϕ), which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption p ≥ n where p is the characteristic of the field) that gives a necessary and sufficient condition for ξ (ϕ) to have a unique irreducible constituent.

Research paper thumbnail of A supercharacter theory for involutive algebra groups

Journal of Algebra, 2015

ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, ... more ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, the algebra group P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If mathscrJ\mathscr{J}mathscrJ is endowed with an involution widehatvarsigma\widehat{\varsigma}widehatvarsigma, then widehatvarsigma\widehat{\varsigma}widehatvarsigma naturally defines a group automorphism of P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ, and we may consider the fixed point subgroup CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1. Assuming that Bbbk\BbbkBbbk has odd characteristic ppp, we use the standard supercharacter theory for PPP to construct a supercharacter theory for CP(widehatvarsigma)C_{P}(\widehat{\varsigma})CP(widehatvarsigma). In particular, we obtain a supercharacter theory for the Sylow ppp-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr\'e and Neto for the special case of the symplectic and orthogonal groups.

Research paper thumbnail of A supercharacter theory for involutive algebra groups

Journal of Algebra, 2015

ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, ... more ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, the algebra group P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If mathscrJ\mathscr{J}mathscrJ is endowed with an involution widehatvarsigma\widehat{\varsigma}widehatvarsigma, then widehatvarsigma\widehat{\varsigma}widehatvarsigma naturally defines a group automorphism of P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ, and we may consider the fixed point subgroup CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1. Assuming that Bbbk\BbbkBbbk has odd characteristic ppp, we use the standard supercharacter theory for PPP to construct a supercharacter theory for CP(widehatvarsigma)C_{P}(\widehat{\varsigma})CP(widehatvarsigma). In particular, we obtain a supercharacter theory for the Sylow ppp-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr\'e and Neto for the special case of the symplectic and orthogonal groups.