arXiv subjects

Gus Lehrer

Publications and source records attributed to Gus Lehrer.

8 recordsLinked to original sources

arXiv2022

Generalised Temperley-Lieb algebras of type $G(r,p,n)$

In an earlier work, we defined a ``generalised Temperley-Lieb algebra'' $TL_{r,1,n}$ corresponding to the imprimitive reflection group $G(r,1,n)$ as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley-Lieb algebra $TL_{r,p,n}$ which corresponds to the complex reflection group $G(r,p,n)$. Our definition identifies $TL_{r,p,n}$ as the fixed-point subalgebra of $TL_{r,1,n}$ under a certain automorphism $\sigma$. We prove the cellularity of $TL_{r,p,n}$ by proving that $\sigma$ induces a special shift automorphism with respect to the cellular structure of $TL_{r,1,n}$. We also give a description of the cell modules of $TL_{r,p,n}$ and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory.

math.RT↗

arXiv2022

Milnor fibre homology complexes

Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as ``the non-crossing algebra'', which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $\widetilde{A}$, which seems to be ``dual'' to the Fomin-Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection between $\widetilde{A}$ and the Orlik-Solomon algebra, in analogy with the fact that the Fomin-Kirillov algebra contains the coinvariant algebra of $W$. This analysis is applied to compute the multiplicities $\langle ρ, H^k(F,\mathbb{C})\rangle_W$ and $\langle ρ, H^k(M,\mathbb{C})\rangle_W$, where $M$ and $F$ are respectively the hyperplane complement and Milnor fibre associated to $W$ and $ρ$ is a representation of $W$.

math.RT↗

arXiv2022

Generalised Temperley-Lieb algebras of type $G(r,1,n)$

In this paper, we define a quotient of the cyclotomic Hecke algebra of type $G(r,1,n)$ as a generalisation of the Temperley-Lieb algebras of type $A$ and $B$. We establish a graded cellular structure for the generalised Temperley-Lieb algebra and, using the technology of $KLR$ algebras, determine the corresponding decomposition matrix.

math.RT↗

arXiv2019

Schur-Weyl duality for certain infinite dimensional $\rm{U}_q(\mathfrak{sl}_2)$-modules

Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show that for any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $\rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $\rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.

math.RT↗

arXiv2013

On the cohomology of the Milnor fibre of a hyperplane arrangement

We investigate the cohomology of the Milnor fibre of a reflection arrangement as a module for the group $Γ$ generated by the reflections, together with the cyclic monodromy. Although we succeed completely only for unitary reflection groups of rank two, we establish some general results which relate the isotypic componenents of the monodromy on the cohomology, to the Hodge structure and to the cohomology degree. Using eigenspace theory for reflection groups, we prove some sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler characteristic, such as the Hodge-Deligne polynomials. We also use monodromy eigenspaces to determine the spectrum in some cases, which in turn throws light on the Hodge structure of the cohomology. These methods enable us to compute the complete story, including the representation of $Γ$ on the Hodge components in each cohomology degree, for some groups of low rank.

math.AG↗

arXiv2011

Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements

We investigate the interplay between the monodromy and the Deligne mixed Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of hyperplane arrangement Milnor fibers, we obtain a new result on the possible weights. For line arrangements, we prove in a new way the fact due to Budur and Saito that the spectrum is determined by the weak combinatorial data, and show that such a result fails for the Hodge-Deligne polynomials.

math.AG↗

arXiv2008

The equivariant Euler characteristic of real Coxeter toric varieties

Let $W$ be a Weyl group, and let $\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\CT_W(\R)$ is a smooth, connected, compact manifold with a $W$-action. We give a formula for the equivariant Euler characteristic of $\CT_W(\R)$ as a generalised character of $W$. In type $A_{n-1}$ for $n$ odd, one obtains a generalised character of $\Sym_n$ whose degree is (up to sign) the $n^{\text{th}}$ Euler number.

math.RT↗

arXiv2006

Twisted invariant theory for reflection groups

Let $G$ be a reflection group acting on a vector space $V$ and let $γ$ be an automorphism of $V$ normalising $G$. We study how $γ$ acts on invariants and covariants (for various representations) of $G$, and properties of its eigenspaces.

math.GR↗