Quantum groups and functional relations for lower rank

Crystal bases and combinatorics of infinite rank quantum groups

The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independant of the infinite root system considered. Although the irreducible modules obtained in this way are not of highest weight, they admit a crystal basis and a canonical basis. This permits in particular to obtain for each familly of classic...

Dimension and Rank for Mapping Class Groups

We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG has quasi-flats of dimension N if and only if it has a rank N free abelian sub...

COUNTING DISTINCT FUZZY SUBGROUPS OF SOME RANK-3 ABELIAN GROUPS

In this paper we classify fuzzy subgroups of a rank-3 abelian group $G = mathbb{Z}_{p^n} + mathbb{Z}_p + mathbb{Z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorp...

Structured Lower Rank Approximation

Rank 3 Groups and Biplanes

Let G be a primitive rank 3 permutation group on a set X in which r(x) is a nontrivial G,-orbit, with II = I X 1, u = I I’(X)]. Tsuzuku [27] showed that, if G, acts as the symmetric group on r(x), then (v, n) = (2, 5), (3, lo), (5, 16), or (7, 50); he determined the possible groups in each case. Bannai [2] obtained essentially the same result under the assumption G, is 4-transitive on F(x). (Of...