arXiv subjects

Pham Huu Tiep

Publications and source records attributed to Pham Huu Tiep.

78 records · Page 5Linked to original sources

arXiv2008

Representations of the general linear groups which are irreducible over subgroups

We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\dar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.

math.RT↗

arXiv2008

Representations of finite special linear groups in non-defining characteristic

We determine precisely the number of irreducible summands of an irreducible cross characteristic representation of $GL_{n}(q)$ on restriction to $SL_{n}(q)$. Combined with a recent result of C. Bonnafe, this yields a canonical labeling for irreducible $\ell$-modular representations of $SL_{n}(q)$, where $(\ell,q)=1$. As an application, we classify for the first time complex representations of $SL_{n}(q)$ whose reductions modulo $\ell$ are irreducible.

math.RT↗

arXiv2008

Hall-Higman type theorems for semisimple elements of finite classical groups

We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.

math.RT↗

arXiv2008

Symmetric powers and a problem of Kollar and Larsen

We prove a conjecture of Kollar and Larsen on Zariski closed subgroups of $GL(V)$ which act irreducibly on some symmetric power $Sym^{k}(V)$ with $k \geq 4$. This conjecture has interesting implications, in particular on the holonomy group of a stable vector bundle on a smooth projective variety, as shown by the recent work of Balaji and Kollar.

math.RT↗

arXiv2005

Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.

quant-ph↗

arXiv2005

Decompositions of Small Tensor Powers and Larsen's Conjecture

We classify all pairs (G,V) with G a closed subgroup in a classical group with natural module V over the complex numbers such that G has the same composition factors on the kth tensor power of V, for a fixed (small) k. In particular, we prove Larsen's conjecture stating that for dim(V) > 6 and k = 4, there are no such G aside from those containing the derived subgroup of the classical group. We also find all the examples where this fails for dim(V) < 7. As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz's recent works on monodromy groups attached to Lefschetz pencils and to character sums over finite fields. Modular versions of these conjectures are also studied, with a particular application to random generation in finite groups of Lie type.

math.GR↗