Reducing of representationsSuppose that we have a set of n-dimensional matrices, A, B,C, ... , which form a representation of a group. These n-Dmatrices themselves constitute a matrix group I= (A, B,..].If we make the same similarity transformation on eachmatrix, we obtain a newset ofmatrices, namelyA'= X-IAX, B'= X-'BX,C'= X-'CX, ...I'=(A', B',C',..that forms a new matrix group: F' is also a representation of the group!
Reducing of representations • Suppose that we have a set of n-dimensional matrices, A, B, C, . , which form a representation of a group. These n-D matrices themselves constitute a matrix group = {A，B,.}. • If we make the same similarity transformation on each matrix, we obtain a new set of matrices，namely, A' ， B' ，C' ,. • is also a representation of the group! that forms a new matrix group: , . 1 1 1 C X CX A X AX B X BX ' ' ， '
It is provable that if any of the matrix (e.g., A') in I' is ablock-factored matrix, then all other matrices (e.g., B,c',...)inrare also blocked-factored000[c]0[B, ]00[4]000[4.]0[B, ] 0[c,] 0000000A':B':[c,] [4, ][B,] 000000000[c.][B ]000000000Ain which A,A2,A3... are n,n2,n3...-order submatrices with n =ng + n2 + n3 +.These n-order matrices can be simply expressed asA'= A,④A2 ④A ④...,B'= B,④B2 ④ B3 ④ ...,C'= C,④C, ④ C 甲...,(Directsumofsubmatrices!
' , ' , ' ，. 4 3 2 1 4 3 2 1 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C C C C C B B B B B A A A A A in which A1 ,A2 ,A3. are n1 ,n2 ,n3.-order submatrices with n = n1 + n2 + n3 + . • It is provable that if any of the matrix (e.g., A ) in is a block-factored matrix, then all other matrices (e.g., B,C ,.) in are also blocked-factored. A= A1A2 A3 ., B= B1B2 B3 ., C= C1C2 C3 ., . • These n-order matrices can be simply expressed as (Direct sum of submatrices！）
It is also provable that the various sets of submatricesT,={A,B1,C....], T2-{A2,B2,C2..], T3-{A3,B3,C3...], ...,are in themselves representations of the groupWe then call the set of matrices I={A,B,C, ...} a reduciblerepresentation of the group, which breaks up into a directsum of the representations, i.e., I = T, ④ T, ④ T3 @ ...If it is not possibleto find a similarity transformationtoreduce a representation in the above manner, therepresentationissaidtobeirreducibleThe irreducible representations of apoint group are mostlycountableandoffundamentalimportance!
• It is also provable that the various sets of submatrices, T1={A1 ,B1 ,C1.}, T2={A2 ,B2 ,C2.}, T3={A3 ,B3 ,C3.}, ., are in themselves representations of the group. • We then call the set of matrices ={A,B,C, .} a reducible representation of the group, which breaks up into a direct sum of the representations, i.e., = T1 T2 T3 . • If it is not possible to find a similarity transformation to reduce a representation in the above manner, the representation is said to be irreducible. • The irreducible representations of a point group are mostly countable and of fundamental importance!
Example: C2vIs this 3-D Rep. reducible?Yes. These matrices are block-factored!EC2OxzOyz00000000一L0000000xyz000000000Y1RedicedoDmatriceirreduciblerepresentation1[1] [-1] [1] [-1]1-1-1X1-11] [-1] [-11 [1]-11S1111=Z[1][1][1][1]Fxz=F ④, ④The 3-D rep. is reduced to 3 1-D rep
Example: C2v Is this 3-D Rep. reducible? xyz xyz =x y z Yes. These matrices are block-factored! The 3-D rep. is reduced to 3 1-D rep
(symm. ops.)Point group RR={RA, Rp, Rc,...]Exerted on any set of bases(e.g., AO's, MO's, vectors, rotations etc.)Amatrix group, I = {A, B, C, ...}(a matrix rep. of group R, dimension = order of the matrix)Similarity transformations (reducing of a representation!A block-factored matrix group, I' ={A, B', C', ...(A'=A,④A,④..., B"=B,B,④..., C' =C,④C,@... ....)and F ={Aj,Bi,C1....] , I, ={A2,B2,C2....] ...&=I④④Direct sum of irreducible representations!
Point group R R={RA，RB，RC，.} A matrix group, = {A，B，C，.} (a matrix rep. of group R, dimension = order of the matrix) Exerted on any set of bases （e.g., AOs, MOs, vectors, rotations etc.) Similarity transformations (reducing of a representation!) Direct sum of irreducible representations! (symm. ops.) A block-factored matrix group, = {A ，B ，C ，.} (A = A1 A2 ., B = B1 B2 ., C = C1 C2 . ,.) and 1 ={A1 ,B1 ,C1 ,.} , 2 ={A2 ,B2 ,C2 ,.} . & = 1 2