5 | | Only if the matrix is representing a linear map and the tuple represents a vector space element there is an assumption on both being represented in the same basis. When you now represent a Coxeter group element as a matrix you make a choice of basis (which might be the root basis, its dual, some linearly independent vectors of an "ambient space", or any other) This is now what you stick this matrix to, but when you do {{{w*v}}}, one does not specify the matrix that represents {{{w}}} in some basis, so this operation is ambiguous, as it didn't specify on which space {{{w}}} acts here (i.e., it is not clear in which basis you represent {{{w}}} as a matrix. |

| 5 | Only if the matrix is representing a linear map and the tuple represents a vector space element there is an assumption on both being represented in the same basis. When you now represent a Coxeter group element as a matrix you make a choice of basis (which might be the root basis, its dual, some linearly independent vectors of an "ambient space", or any other) This is now what you stick this matrix to, but when you do {{{w*v}}}, one does not specify the matrix that represents {{{w}}} in some basis, so this operation is ambiguous, as it didn't specify on which space {{{w}}} acts here (i.e., it is not clear in which basis you represent {{{w}}} as a matrix for that action. |