Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-... more Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is "uniform" over good primes p > 3, but differs for the bad prime p = 3. We also explain how it has been applied for all groups of rank 4 or less.
Here we study the automorphism groups of 1-designs constructed from finite nonabelian simple grou... more Here we study the automorphism groups of 1-designs constructed from finite nonabelian simple groups by using two methods presented in Moori (Information Security, Coding Theory and Related Combinatorics, 2011). We obtain some general results for both and improve one of these methods. In an application to the sporadic Mathieu groups $M_n$, we are able to retrieve the Steiner systems $S(t,t+3,n)$ where $(n,t)\in\{(22,3),(23,4),(24,5)\}$.
Here we construct and count all ordinary irreducible characters of Sylow p-subgroups of the Stein... more Here we construct and count all ordinary irreducible characters of Sylow p-subgroups of the Steinberg triality groups $^3D_4(p^{3m})$.
Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We ... more Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We describe a construction of all complex irreducible characters of U(q) and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in (q−1) with nonnegative integer coefficients.
Let Un(q) denote the upper triangular group of degree n over the finite field Fq with q elements.... more Let Un(q) denote the upper triangular group of degree n over the finite field Fq with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form U k (q)∩ w U k (q) where w is a monomial matrix in GL k (q) for some k < n.
Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of th... more Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of the irreducible characters of the Sylow $p$-subgroup of $GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed that this is true for a Sylow $p$-subgroup of a classical group defined over $\F_q$ if and only if $p$ is odd. For the classical groups of Lie type $B$, $C$ and $D$ the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow $p$-subgroups of the Chevalley groups $D_4(q)$ with $q=2^f$ of degree $q^3/2$. Then we use an analogous construction for $E_6(q)$ with $q=3^f$ to obtain characters of degree $q^7/3$, and for $E_8(q)$ with $q=5^f$ to obtain characters of degree $q^{16}/5.$ This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type $E$ in terms of the representation theory of the Sylow $p$-subgroup.
Let Un(q) be the upper triangular group of degree n over the finite field Fq with q elements. In ... more Let Un(q) be the upper triangular group of degree n over the finite field Fq with q elements. In this paper, we present constructions of large degree (complex) irreducible representations of Un(q) where n⩾7, and then determine the number of irreducible representations of largest, second largest and third largest degrees.
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.
Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-... more Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is "uniform" over good primes p > 3, but differs for the bad prime p = 3. We also explain how it has been applied for all groups of rank 4 or less.
Here we study the automorphism groups of 1-designs constructed from finite nonabelian simple grou... more Here we study the automorphism groups of 1-designs constructed from finite nonabelian simple groups by using two methods presented in Moori (Information Security, Coding Theory and Related Combinatorics, 2011). We obtain some general results for both and improve one of these methods. In an application to the sporadic Mathieu groups $M_n$, we are able to retrieve the Steiner systems $S(t,t+3,n)$ where $(n,t)\in\{(22,3),(23,4),(24,5)\}$.
Here we construct and count all ordinary irreducible characters of Sylow p-subgroups of the Stein... more Here we construct and count all ordinary irreducible characters of Sylow p-subgroups of the Steinberg triality groups $^3D_4(p^{3m})$.
Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We ... more Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We describe a construction of all complex irreducible characters of U(q) and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in (q−1) with nonnegative integer coefficients.
Let Un(q) denote the upper triangular group of degree n over the finite field Fq with q elements.... more Let Un(q) denote the upper triangular group of degree n over the finite field Fq with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form U k (q)∩ w U k (q) where w is a monomial matrix in GL k (q) for some k < n.
Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of th... more Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of the irreducible characters of the Sylow $p$-subgroup of $GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed that this is true for a Sylow $p$-subgroup of a classical group defined over $\F_q$ if and only if $p$ is odd. For the classical groups of Lie type $B$, $C$ and $D$ the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow $p$-subgroups of the Chevalley groups $D_4(q)$ with $q=2^f$ of degree $q^3/2$. Then we use an analogous construction for $E_6(q)$ with $q=3^f$ to obtain characters of degree $q^7/3$, and for $E_8(q)$ with $q=5^f$ to obtain characters of degree $q^{16}/5.$ This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type $E$ in terms of the representation theory of the Sylow $p$-subgroup.
Let Un(q) be the upper triangular group of degree n over the finite field Fq with q elements. In ... more Let Un(q) be the upper triangular group of degree n over the finite field Fq with q elements. In this paper, we present constructions of large degree (complex) irreducible representations of Un(q) where n⩾7, and then determine the number of irreducible representations of largest, second largest and third largest degrees.
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.
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