The Alexander polynomial as quantum invariant of links

In these notes we collect some results about finite dimensional representations of $U_q(\mathfrak{gl}(1|1))$ and related invariants of framed tangles which are well-known to experts but difficult to find in the literature. In particular, we give an explicit description of the ribbon structure on the category of finite dimensional $U_q(\mathfrak{gl}(1|1))$-representation and we use it to construct the corresponding quantum invariant of framed tangles. We explain in detail why this invariant vanishes on closed links and how one can modify the construction to get a nonzero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of $U_q(\mathfrak{gl}(1|1))$.Comment: This is a corrected and revised version, to be published on Arkiv f\"or Matematik. This is part of the author's PhD Thesi

A diagram algebra for Soergel modules corresponding to smooth Schubert varieties

Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module categories are equivalent to the subquotient categories of the BGG category $\mathcal{O}(\mathfrak{gl}_n)$ which show up in categorification of $\mathfrak{gl}(1|1)$-representations. We construct diagrammatically the graded cellular structure and the properly stratified structure of these algebras.Comment: A previous version of this paper was originally included as Part 3 in arXiv:1305.6162, which now has been replaced. This paper contains many corrections and improvements. This is part of the author's PhD thesi

Walled Brauer algebras as idempotent truncations of level 2 cyclotomic quotients

We realize (via an explicit isomorphism) the walled Brauer algebra for an arbitrary integral parameter delta as an idempotent truncation of a level two cyclotomic degenerate affine walled Brauer algebra. The latter arises naturally in Lie theory as the endomorphism ring of so-called mixed tensor products, i.e. of a parabolic Verma module tensored with some copies of the natural representation and its dual. This provides us a method to construct central elements in the walled Brauer algebras and can be applied to establish the Koszulity of the walled Brauer algebra if delta is not zero.Comment: Some correction

La escritura cotidiana, negazione dell’epigrafia?

Le scritture quotidiane, di per sé effimere per tecnica esecutiva, per oggetto del messaggio contenutovi, per finalità limitate, risultano prive degli espedienti tecnici, delle forme di visibilità, dell’intenzione mirata di preservare la più duratura validità, rinunciando di fatto alle caratteristiche più proprie delle iscrizioni epigrafiche «vere».Les écritures quotidiennes sont par elles-mêmes éphémères par les téchniques d’exécution, par leurs contenus, par leurs buts trop brefs. Comme elles n’ont pas suffisamment ni des expédients téchniques, ni de l’espoir de visibilité, ni la volonté de s’assurer une durée convenable, elles renoncent de fait aux fondamentaux caractéristiques des inscriptions épigraphiques réelles

La comunicazione epigrafica e l’epigrafia comunicata

Si definiscono le caratteristiche fondamentali della comunicazione epigrafica, che condizionano la presentazione museale delle iscrizioni, e si considerano gli esempi più nuovi dei lapidari dell’Italia Settentrionale. ______________________________________________ Hier werden die Grundeigenschaften der epigraphischen Kommunikation definiert, auf denen die Präsentation der Inschriften im Museum basiert. Zudem werden die neuesten Beispiele norditalienischer Inschriftensammlungen betrachtet