Posets of Non-Crossing Partitions of Type B and Applications

Abstract

The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.