arXiv subjects

Lacrimioara Iancu

Publications and source records attributed to Lacrimioara Iancu.

5 recordsLinked to original sources

arXiv2014

Coxeter groups and automorphisms

Let $(W,S)$ be a Coxeter system and $Γ$ be a group of automorphisms of $W$ such that $γ(S)=S$ for all $γ\in Γ$. Then it is known that the group of fixed points $W^Γ$ is again a Coxeter group with a canonically defined set of generators. The usual proofs of this fact rely on the reflection representation of $W$. Here, we give a proof which only uses the combinatorics of reduced expressions in $W$. As a by-product, this shows that the length function on $W$ restricts to a weight function on $W^Γ$.

math.RT↗

arXiv2012

Ordering Lusztig's families in type $B_n$

Let $W$ be a finite Coxeter group and $L$ be a weight function on $W$ in the sense of Lusztig. We have recently introduced a pre-order relation $\preceq_L$ on the set of irreducible characters of $W$ which extends Lusztig's definition of "families" and which, conjecturally, corresponds to the ordering given by Kazhdan--Lusztig cells. Here, we give an explicit description of $\preceq_L$ for $W$ of type $B_n$ and any $L$. (All other cases are known from previous work.) This crucially relies on some new combinatorial constructions around Lusztig's "symbols". Combined with previous work, we deduce general compatibility results between $\preceq_L$ and Lusztig's $\ba$-function, valid for any $W,L$.

math.RT↗

arXiv2007

Specht modules and Kazhdan--Lusztig cells in type $B_n$

Dipper, James and Murphy generalized the classical Specht module theory to Hecke algebras of type $B_n$. On the other hand, for any choice of a monomial order on the parameters in type $B_n$, we obtain corresponding Kazhdan--Lusztig cell modules. In this paper, we show that the Specht modules are naturally equivalent to the Kazhdan--Lusztig cell modules {\em if} we choose the dominance order on the parameters, as in the ``asymptotic case'' studied by Bonnafé and the second named author. We also give examples which show that such an equivalence does not hold for other choices of monomial orders.

math.RT↗

arXiv2007

On domino insertion and Kazhdan--Lusztig cells in type $B_n$

Based on empirical evidence obtained using the {\sf CHEVIE} computer algebra system, we present a series of conjectures concerning the combinatorial description of the Kazhdan--Lusztig cells for type $B_n$ with unequal parameters. These conjectures form a far-reaching extension of the results of Bonnafé and Iancu obtained earlier in the so-called ``asymptotic case''. We give some partial results in support of our conjectures.

math.RT↗

arXiv2005

Lusztig's $a$-function in type $B_n$ in the asymptotic case

In this paper, we study Lusztig's $a$-function for a Coxeter group with unequal parameters. We determine that function explicitly in the ``asymptotic case'' in type $B_n$, where the left cells have been determined in terms of a generalized Robinson--Schensted correspondence by Bonnafé and the second author. As a consequence, we can also show that all of Lusztig's conjectural properties (P1)--(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the ``leading matrix coefficients'' introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori--Hecke algebra $\bH_n$ in terms of the Dipper--james--Murphy basis of $\bH_n$.

math.RT↗