Faithful representation

In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G {displaystyle G} on a vector space V {displaystyle V} is a linear representation in which different elements g {displaystyle g} of G {displaystyle G} are represented by distinct linear mappings ρ ( g ) {displaystyle ho (g)} . In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G {displaystyle G} on a vector space V {displaystyle V} is a linear representation in which different elements g {displaystyle g} of G {displaystyle G} are represented by distinct linear mappings ρ ( g ) {displaystyle ho (g)} . In more abstract language, this means that the group homomorphism ρ : G → G L ( V ) {displaystyle ho :G o GL(V)} is injective (or one-to-one). Caveat: While representations of G {displaystyle G} over a field K {displaystyle K} are de facto the same as K [ G ] {displaystyle K} -modules (with K [ G ] {displaystyle K} denoting the group algebra of the group G {displaystyle G} ), a faithful representation of G {displaystyle G} is not necessarily a faithful module for the group algebra. In fact each faithful K [ G ] {displaystyle K} -module is a faithful representation of G {displaystyle G} , but the converse does not hold. Consider for example the natural representation of the symmetric group S n {displaystyle S_{n}} in n {displaystyle n} dimensions by permutation matrices, which is certainly faithful. Here the order of the group is n {displaystyle n} ! while the n × n {displaystyle n imes n} matrices form a vector space of dimension n 2 {displaystyle n^{2}} . As soon as n {displaystyle n} is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since 24 > 16 {displaystyle 24>16} ); this relation means that the module for the group algebra is not faithful. A representation V {displaystyle V} of a finite group G {displaystyle G} over an algebraically closed field K {displaystyle K} of characteristic zero is faithful (as a representation) if and only if every irreducible representation of G {displaystyle G} occurs as a subrepresentation of S n V {displaystyle S^{n}V} (the n {displaystyle n} -th symmetric power of the representation V {displaystyle V} ) for a sufficiently high n {displaystyle n} . Also, V {displaystyle V} is faithful (as a representation) if and only if every irreducible representation of G {displaystyle G} occurs as a subrepresentation of (the n {displaystyle n} -th tensor power of the representation V {displaystyle V} ) for a sufficiently high n {displaystyle n} . Hazewinkel, Michiel, ed. (2001) , 'faithful representation', Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:''''''''''''}.mw-parser-output .citation .cs1-lock-free a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}