associated to the spin structure P {\displaystyle {\mathbf {P} }} via the spin representation κ : S p i n ( n ) → U ( Δ n ) , {\displaystyle \kappa \colon {\mathrm {Spin} }(n)\to {\mathrm {U} }(\Delta _{n}),\,} where U ( W ) {\displaystyle {\mathrm {U} }({\mathbf {W} })\,} denotes the group of unitary operators acting on a Hilbert space W . {\displaystyle {\mathbf {W} }.\,} It is worth noting that the spin representation κ {\displaystyle \kappa } is a faithful and unitary representation of the group S p i n ( n ) {\displaystyle {\mathrm {Spin} }(n)} .[2]