  Reconfigurable intelligent surfaces (RISs) have emerged as a spectrum- and
energy-efficient technology to enhance the coverage of wireless communications
within the upcoming 6G networks. Recently, novel extensions of this technology,
referred to as multi-sector beyond diagonal RIS (BD-RIS), have been proposed,
where the configurable elements are divided into $L$ sectors $(L \geq 2)$ and
arranged as a polygon prism, with each sector covering $1/L$ space. This paper
presents a performance analysis of a multi-user communication system assisted
by a multi-sector BD-RIS operating in time-switching (TS) mode. Specifically,
we derive closed-form expressions for the moment-generating function (MGF),
probability density function (PDF), and cumulative density function (CDF) of
the signal-to-noise ratio (SNR) per user. Furthermore, closed-form expressions
for the outage probability, achievable spectral and energy efficiency, symbol
error probability, and diversity order for the proposed system model are
derived. Moreover, a comparison is performed with the simultaneously
transmitting and reflecting (STAR)-RISs, a special case of multi-sector BD-RIS
with two sectors. Our analysis shows that for a fixed number of elements,
increasing the sectors improves outage performance at the expense of reduced
diversity order compared to STAR-RIS. This trade-off is influenced by the
Rician factors of the cascaded channel and the number of configurable elements
per sector. However, this superiority in slope is observed at outage
probability values below $10^{-5}$, which remains below practical operating
ranges of communication systems. Additionally, simulations are provided to
validate the accuracy of our theoretical analyses showing a notable $182\%$
increase in spectral efficiency and a $238\%$ increase in energy efficiency
when transitioning from a 2-sector to a 6-sector configuration.
