Narrow Results

In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of ${\mathrm{\text{SL}}}_3(k)$ and ${\mathrm{\text{SL}}}_4(k)$ where $k$ is a finite field of even characteristic. We also provide some generalized results for ${\mathrm{\text{SL}}}_n(k)$. These results, together with our previous work that determined similar orbits when $k$ is a finite field of odd characteristic, complete the classification of the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of ${\mathrm{\text{SL}}}_3(k)$ and ${\mathrm{\text{SL}}}_4(k)$ for any finite field $k$ of any characteristic.

Let $R$ be a commutative local ring with residue field $\overline{R}$. Suppose $n>1$ is an even natural number, $\overline{R}$ contains at least $n+3$ elements and $q(x)=x^2-cx+1$ is a quadratic polynomial in $R[x]$. The main goal of this paper is to prove that the covering number ${\mathrm{\text{SL}}}_n(R)$ with respect to the set of $q(x)$-quadratic matrices in ${\mathrm{\text{SL}}}_n(R)$ is equal to or less than $5$. Some corollaries are also presented.

We study the distance on the Bruhat–Tits building of the group ${\mathrm{\text{SL}}}_d({\mathbb{Q}}_p)$ (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with combinatorial meanings. In Theorem 1, we give an explicit formula for the graph distance $\delta (\alpha ,\beta )$ of two vertices $\alpha$ and $\beta$ (without having to specify their common apartment). Our main result, Theorem 2, then extends the distance formula to a formula for the smallest total distance of a vertex from a given finite set of vertices. In the appendix we consider the case of ${\mathrm{\text{SL}}}_2({\mathbb{Q}}_p)$ and give a formula for the number of edges shared by two given apartments.

Let K be a principal ideal domain, and A_n, with n ≥ 3, be a finitely generated torsion-free abelian group of rank n. Let Ω be a finite subset of KA_n\{0} and U(KA_n) the group of units of KA_n. For a multiplicative monoid P generated by U(KA_n) and Ω, we prove that any generating set for contains infinitely many elements not in . Furthermore, we present a way of constructing elements of not in for n ≥ 3. In the case where K is not a field the aforementioned results hold for n ≥ 2.

We determine the integers a, b ≥ 1 and the prime powers q for which the word map w(x, y) = x^ay^b is surjective on the group PSL(2, q) (and SL(2, q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2, q) (and SL(2, q)). Our proof is based on the investigation of the trace map of positive words.

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an ${\mathrm{\text{SU}}}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t=1$. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most $10$. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary ${\mathrm{\text{SL}}}_2$- or ${\mathrm{\text{SU}}}_2$-module as well as its first three Laurent coefficients.

Let $V$ be a finite-dimensional representation of the group ${SL}_2$ of $2\times 2$ matrices with complex coefficients and determinant one. Let $R=ℂ{[V]}^{{SL}_2}$ be the algebra of ${SL}_2$-invariant polynomials on $V$. We present a calculation of the Hilbert series ${\mathrm{\text{Hilb}}}_R(t)={\sum }_{n\ge 0}dim(R_n)t^n$ as well as formulas for the first four coefficients of the Laurent expansion of ${\mathrm{\text{Hilb}}}_R(t)$ at $t=1$.

Let $G$ be a group and $h,g\in G$. The 2-tuple $(h,g)$ is said to be an $n$-Engel pair, $n\ge 2$, if $h=[h{,}_{n}g]$, $g=[g{,}_{n}h]$ and $h\ne 1$. Let SL$(2,F)$ be the special linear group of degree $2$ over the field $F$. In this paper, we show that given any field $L$, there is a field extension $F$ of $L$ with $[F:L]\le 6$ such that SL$(2,F)$ has an $n$-Engel pair for some integer $n\ge 4$. We will also show that SL$(2,F)$ has a $5$-Engel pair if $F$ is a field of characteristic $p\equiv \pm 1mod5$.

A strong Gelfand pair $(G,H)$ is a group $G$ together with a subgroup $H$ such that every irreducible character of $H$ induces a multiplicity-free character of $G$. We classify the strong Gelfand pairs of the special linear groups $\mathrm{\text{SL}}(2,p)$, where $p$ is a prime.

In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of ${\mathrm{\text{SL}}}_3(k)$ and ${\mathrm{\text{SL}}}_4(k)$ where $k$ is a finite field of odd characteristic. We provide some generalized results for ${\mathrm{\text{SL}}}_n(k)$.

The blockwise Alperin weight conjecture assets that for any finite group G and any prime l, the number of the Brauer characters in an l-block B equals the number of the G-conjugacy classes of l-weights belonging to B. Recently, the inductive blockwise Alperin weight condition has been introduced such that the blockwise Alperin weight conjecture holds if all non-abelian simple groups satisfy these conditions. We will verify the inductive blockwise Alperin weight condition for the finite simple groups PSL(3, q) in this paper.

Unlike the traditional independent component analysis (ICA) algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices, in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening. The new algorithms are based on the special linear group SL(n). In order to achieve our target, we first provide a representation theory for any matrix in SL(n), which only simply uses the product of multiple exponentials of traceless matrices. Based on the matrix representation theory, two novel ICA algorithms are developed along with simple analysis on their equilibrium points. Moreover, we apply our methods to the classical problem of signal separation. The experimental results indicate that the superior convergence of our proposed algorithms, which can be expected as two viable alternatives to the ICA algorithms available in publications.

The purpose of this paper is to articulate an observation that many interesting types of wavelets (or coherent states) arise from group representations which are not square integrable or vacuum vectors which are not admissible.