2009/04/21 by Karola Mészáros, Karola Meszaros, Meszaros, Karola
Mathematics · #05E15 #16S99 #51M25 #52B11 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #math.CO #msc:05E15 #msc:16S99 #msc:51M25 #msc:52B11
paper · pdf · doi:10.48550/arxiv.0904.3339
34 pages, 3 figures; Sections 12, 13, 14 are added, where the type D_n bracket algebra is studied; techniques from noncommutative Groebner bases are used
openalex publication_date 2009/04/21 · arxiv created 2009/09/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The type Cn full root polytope is the convex hull in Rn of the origin and the points ei-ej, ei+ej, 2ek for 1 <= i < j <= n, k ∈ [n]. Given a graph G, with edges labeled positive or negative, associate to each edge e of G a vector v(e) which is ei-ej if e=(i, j), i < j, is labeled negative and ei+ej if it is labeled positive. For such a signed graph G, the associated root polytope P(G) is the intersection of the full root polytope with the cone generated by the vectors v(e), for edges e in G. The reduced forms of a certain monomial m[G] in commuting variables xij, yij, zk under reductions derived from the relations of a bracket algebra of type Cn, can be interpreted as triangulations of P(G). Using these triangulations, the volume of P(G) can be calculated. If we allow variables to commute only when all their indices are distinct, then we prove that the reduced form of m[G], for "good" graphs G, is unique and yields a canonical triangulation of P(G) in which each simplex corresponds to a noncrossing alternating graph in a type C sense. A special case of our results proves a conjecture of A. N. Kirillov about the uniqueness of the reduced form of a Coxeter type element in the bracket algebra of type Cn. We also study the bracket algebra of type Dn and show that a family of monomials has unique reduced forms in it. A special case of our results proves a conjecture of A. N. Kirillov about the uniqueness of the reduced form of a Coxeter type element in the bracket algebra of type Dn.