  In this paper, we study the general rogue wave solutions and their patterns
in the vector (or $M$-component) nonlinear Schr\"{o}dinger (NLS) equation. By
applying the Kadomtsev-Petviashvili hierarchy reduction method, we derived an
explicit solution for the rogue wave expressed by $\tau$ functions that are
determinants of $K\times K$ block matrices ($K=1,2,\cdots, M$) with an index
jump of $M+1$. Patterns of the rogue waves for $M=3,4$ and $K=1$ are thoroughly
investigated. We find that when a specific internal parameter is large enough,
the wave patterns are linked to the root structures of generalized
Wronskian-Hermite polynomial hierarchy in contrast with rogue wave patterns of
the scalar NLS equation, the Manakov system and many others. Moreover, the
generalized Wronskian-Hermite polynomial hierarchy includes the
Yablonskii-Vorob'ev polynomial hierarchy and Okamoto polynomial hierarchies as
special cases, which have been used to describe the rogue wave patterns of the
scalar NLS equation and the Manakov system, respectively. As a result, we
extend the most recent results by Yang {\it et al.} for the scalar NLS equation
and the Manakov system. It is noted that the case $M=3$ displays a new feature
different from the previous results. The predicted rogue wave patterns are
compared with the ones of the true solutions for both cases of $M=3,4$. An
excellent agreement is achieved.
