In this paper‎, ‎we first present some properties of lower and upper central series of pair of groups‎. ‎Then the notion of $n$-isoclinism for the classification of pairs of groups is introduced‎, ‎and some of the structural properties of the created classes are proved‎. ‎Moreover some interesting theorems such as Baer Theorem‎, ‎Bioch Theorem‎, ‎Hirsh Theorem for pair of groups are generalized...

We study the existence of symplectic resolutions of quotient singularities V/G, where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form KoS2 where K < SL2(C), for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for dimV 6= 4, we...

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of...

Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its re...

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ irreducible smooth representations within Arthur-type class. We extend this prediction full class unitarizable representations, exhibiting combinatorial relation must be satisfied for any pair in appears as quotient other. settle...

The classical McKay correspondence for finite subgroups G of SL(2,C) gives a bijection between isomorphism classes of nontrivial irreducible representations of G and irreducible components of the exceptional divisor in the minimal resolution of the quotient singularity A2C/G. Over non algebraically closed fields K there may exist representations irreducible over K which split over K. The same i...

Let X be a Hausdorff quotient of standard space (that is locally compact separable metric space). It shown that the following are equivalent: (i) image an irreducible map from space; (ii) has sequentially dense subset satisfying two technical conditions involving double sequences; (iii) whenever q : Y → , restriction ⁎ | V onto (where pure derived and closure set singleton fibres ). The proof u...

To understand the larger context, first note that Casselman’s subrepresentation theorem asserts that any irreducible representation [1] π of G is a subrepresentation of some one of the principal series representations Is defined below. [2] Further, the quotient Is/π (or a further quotient that is irreducible) again imbeds in some Is′ with another parameter value s′. Thus, any irreducible π appe...

Let F be a locally compact non-Archimedean field, and let B/F be a division algebra of dimension 4. The JacquetLanglands correspondence provides a bijection between smooth irreducible representations π′ of B× of dimension > 1 and irreducible cuspidal representations of GL2(F ). We present a new construction of this bijection in which the preservation of epsilon factors is automatic. This is don...

If F is a local non-Archimedean field, then every irreducible admissible representation π of GL(r, F) is a quotient of a representation ξ induced by tempered ones. We show that £ has a Whittaker model, even though it may fail to be irreducible.