Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn action of the Coxeter group $BC_n$ on maps on surfaces, Lagrangian matroids and their representations
69
0
0.0
(
0
)
Authors
Goran Malic
Ask ChatGPT about the research
No Arabic abstract
For a map $mathcal M$ cellularly embedded on a connected and closed orientable surface, the bases of its Lagrangian (also known as delta-) matroid $Delta(mathcal M)$ correspond to the bases of a Lagrangian subspace $L$ of the standard orthogonal space $mathbb{Q}^Eoplusmathbb{Q}^{E^*}$, where $E$ and $E^*$ are the edge-sets of $mathcal M$ and its dual map. The Lagrangian subspace $L$ is said to be a representation of both $mathcal M$ and $Delta(mathcal M)$. Furthermore, the bases of $Delta(mathcal M)$, when understood as vertices of the hypercube $[-1,1]^n$, induce a polytope $mathbf P(Delta(mathcal M))$ with edges parallel to the root system of type $BC_n$. In this paper we study the action of the Coxeter group $BC_n$ on $mathcal M$, $L$, $Delta(mathcal M)$ and $mathbf P(Delta(mathcal M))$. We also comment on the action of $BC_n$ on $mathcal M$ when $mathcal M$ is understood a dessin denfant.
rate research
Read More
We show that the mapping class group acts properly on the space of maximal representations of the fundamental group of a closed Riemann surface into G when G = Sp(2n,R), SU(n,n), SO*(2n) or Spin(2,n).
For an abelian group $Gamma$, a $Gamma$-labelled graph is a graph whose vertices are labelled by elements of $Gamma$. We prove that a certain collection of edge sets of a $Gamma$-labelled graph forms a delta-matroid, which we call a $Gamma$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by $k$ and Maximum Weight $S$-Tree Packing. We also discuss various properties of $Gamma$-graphic delta-matroids.
Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ fail independently with probability $q in[0,1]$. The emph{all-terminal reliability} of $G$ is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in $q$, and it was conjectured cite{BC1} that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few and far between, and the roots are only barely outside the disk. In this paper we generalize the notion of reliability to simplicial complexes and matroids and investigate when, for small simplicial complexes and matroids, the roots fall inside the closed unit disk.
Given a 3-connected biased graph $Omega$ with a balancing vertex, and with frame matroid $F(Omega)$ nongraphic and 3-connected, we determine all biased graphs $Omega$ with $F(Omega) = F(Omega)$. As a consequence, we show that if $M$ is a 4-connected nongraphic frame matroid represented by a biased graph $Omega$ having a balancing vertex, then $Omega$ essentially uniquely represents $M$. More precisely, all biased graphs representing $M$ are obtained from $Omega$ by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.
In this note we show that if the automorphism group of a normal affine surface $S$ is isomorphic to the automorphism group of a Danielewski surface, then $S$ is isomorphic to a Danielewski surface.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
