The eigenvalues of the quadratic Casimir operator and second-order indices of a simple Lie algebra Howard E Haber Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA Abstract In these notes, we demonstrate how to compute the eigenvalue of the quadratic Casimir operator and the second-order index for an irreducible representation of a simple Lie algebra Explicit results for the fundamental and adjoint representations of su(n), so(n) and sp(n) are given The relation of these results to the dual Coxeter number is clarified Finally, the dependence on the normalization of the Lie algebra generators is discussed I Introduction The reader is assumed to be familiar with Dynkin s techniques for analyzing the simple Lie algebras These methods will be briefly summarized below The material in these notes and further details can be found in refs [ ] The generators of a Lie group G [which constitute a basis for the corresponding Lie algebra g] satisfy the commutation relations [T a, T b ] = f c abt c, a,b,c =,,,d G, () where d G is the dimension of the Lie group G, and there is an implicit sum over repeated indices In eq (), we employ the mathematics convention in which the T a are anti-hermitian generators and the fab c are real structure constants for a compact real Lie algebra The Killing form is defined in terms of a symmetric metric tensor, The inverse of g ab will be denoted by g ab ; that is, g ab = f d acf c bd () g ab g bc = δ c a The adjoint representation consist of d G d G matrices that represent the T a These matrices, which we denote henceforth by F a, are defined by: (F a ) b c = f c ab, (3)
where b and c label the row and column indices of the F a Eq () can then be rewritten as: g ab = Tr(F a F b ) (4) The quadratic Casimir operator, C, is defined by It is easy to prove that C g ab T a T b (5) [C, T a ] = 0, a =,,,d G For a given representation of the Lie algebra g, the generators are represented by d R d R matrices R a By Schur s lemma, any operator that commutes with all the generators of g in an irreducible representation must be a multiple of the identity operator Thus, we shall write: C (R) = g ab R a R b = c R, (6) where is the d R d R identity matrix, and c R is a number that depends only on the representation R The goal of this note is to compute c R for any irreducible representation of a simple Lie group In fact, we can immediately prove the following theorem Theorem : For the adjoint representation (denoted by R = A) of a simple Lie group, c A = Proof: Using the explicit form for the adjoint representation generators given in eq (3), C (A) c e g ab (F a ) c d (F b ) d e = g ab f d ac fe bd = c Aδ e c Multiplying both sides of the above equation by δ c e and we immediately obtain c A = d G c A = g ab f d ac fc bd = gab g ab = d G, and summing over c and e, II Root vectors We choose to work in the Cartan-Weyl basis of g, where the generators consist of {H j, E α }, which satisfy: [H j, H k ] = 0, (7) [H j, E α ] = α j E α, (8) [E α, E α ] = α j H j, α j g jk α k, (9) { N αβ E α+β, if α+β is a root and α+β 0, [E α, E β ] = (0) 0, if α+β is not a root
Here, j =,,,l defines the rank l of the Lie algebra (and corresponds to the maximal number of commuting generators), and the root-vectors are real l- dimensional vectors α = (α, α,, α l )whose components aredefined by eq (8) The set of non-zero root vectors is denoted by Note that α implies that α and kα / if k ± The l l block of the metric tensor is given by: g ij = α α i α j, () and the off-diagonal blocks, g α,j = g j,α = 0 The inverse of g ij (denoted below by g ij )can beused to define inner products of two-vectors that live inthe l-dimensional root vector space, (α, β) = g jk α j β k () It is convenient to choose the normalization of the generators E α of the Cartan-Weyl basis such that g α, α = (3) In this convention, one can show that: N αβ = (α, α)q(p+), N α, β = N α,β, where the integers non-negative p and q are determined by the requirement that β+kα is a root vector for every integer k that satisfies p k q In particular, p q = (β, α) (α, α) (4) Conventionally, one chooses the N αβ to be real We can therefore introduce an ordering of the root vectors by defining α > β if the first non-zero component of α β with respect to some fixed basis is positive One can divide up the roots into two sets: the set of positive roots, denoted by +, and the set of negative roots, denoted by Note that the quadratic Casimir operator can be written in terms of the Cartan-Weyl basis as: C = g ij H i H j + j= α + (E α E α +E α E α ) (5) Finally, we define the simple roots to be a positive root that cannot be expressed as a sum of two other positive roots One can prove that there are precisely l positive roots in a Lie algebra of rank l The set of simple roots is denoted by Π Theorem : If α, β Π and α β, then α β is not a root, and (α, β) 0 (6) 3
Proof: If α β +, then α = (α β) + β shows that α is the sum of two positive roots, which is impossible as α Π Likewise, if β α +, then β = (β α)+α shows that β is the sum of two positive roots, which is impossible as β Π Since α β, it follows that α β is not a root This implies that p = 0 in eq (4), and it follows that (α, β) 0 It is convenient to introduce the l l Cartan matrix A ij, which is defined by: A mn (α m, α n ) (α m, α m ), (7) where m and n label the simple roots Note that A ii = and A ij 0 for i j There is a one-to-one correspondence between the possible Cartan matrices and the Dynkin diagrams that characterize the possible simple Lie groups Given the Cartan matrix, one can compute the inner product of any two simple roots as follows First we note the following result obtained in ref [4], { } (α i, α i ) = k β j A ij, α i Π (8) β + j= where the positive root β has been expressed in terms of the simple roots via β = l j= kβ j α j It then follows from eq (7) that (α i, α j ) = A ij(α i, α i ) We next introduce the Weyl reflection, which acts on a root vector as follows: S i (α) α (α, α i) (α i, α i ) α i, α and α i Π Two immediate properties of S i are: S i (α i ) = α i, (9) (S i (α), β) = (α, S i (β)) (0) Additional properties of the Weyl reflection are summarized by the following theorem Theorem 3: If α + and α α i (for some simple root α i Π), then S i (α) > 0 Moreover, if S i (α) = S i (β), then α = β Proof: [6] Any positive root α + can be written as α = k i α i + j i k j α j, k i,k j 0 Then, S i (α) = α (α, α [ i) (α i, α i ) α i = α i k i + ] k j A ij + k j α j j i j i 4
Since S i (α), it follows that S i (α) is either positive or negative If S i (α) < 0, then we must have k j = 0 for j i, in which case α = α i (ie k i = ) and we recover eq (9) Hence if α α i, it then follows that S i (α) > 0 Next, if S i (α) = S i (β), then α β = κα i, where κ = (α β, α i) (α i, α i ) Inserting α β = κα i into theexpression above yields κα i = κα i, andwe conclude that κ = 0 or α = β One consequence of Theorem 3 is that S i maps the set of positive roots excluding α i into itself, where the map is one-to-one and onto Thus, if we define the Weyl vector δ to be half the sum of the positive roots, δ α, () α + then using eq (9), S i (δ) = S i ( α i + j i α j ) = ( α i + j i α j ) = δ α i () Hence, eq (0) yields, Using eqs (9) and (), it follows that Rearranging the above result then yields: (S i (δ), α i ) = (δ, S i (α i )) (δ α i, α i ) = (δ, α i ) (δ, α i ) (α i, α i ) Finally, we introduce the dual root or co-root of α, = (3) α α (α, α) (4) In terms of the dual root, the Cartan matrix can be defined as A mn = (α m, α n), and the Weyl reflection acts on a root vector as follows: Eq (3) then can be rewritten as: S i (α) = α (α, α i )α i (δ, α i ) = 5
III Irreducible representations and weights In a unitary representation of a simple Lie algebra, the representation matrices of the Cartan-Weyl generators satisfy H j = H j and E α = E α To construct a particular representation, one determines the basis vectors of the representation space, denoted collectively by m These vectors are chosen to be the simultaneous eigenvectors of the commuting Hermitian H j, H j m = m j m The components of the l-dimensional vector m = (m, m,, m l ) are the corresponding eigenvalues of H j The l-dimensional vector space in which the m reside is called the vector space of weight vectors We can formulate a ordering of vectors of the weight space by introducing the rule that m > n if the first non-zero component of m n is positive An important theorem in Lie algebra representation theory states that for a given irreducible representation, the highest weight M is non-degenerate and uniquely fixes the representation Moreover, E α M = 0, for all α + (5) An irreducible representation of a simple Lie algebra can also be specified by their Dynkinlabels Givenaconventionalorderedlistofthesimpleroots,{α, α,, α l }, of g, one can define the integers: n i (M, α i) (α i, α i ) = (M, α i ), i =,,,l (6) One can prove that the n i are non-negative integers Thus, an irreducible representation can be identified by the order pair (n, n,, n l ), where the n i are called the Dynkin labels of the irreducible representation Since M is a vector that lives in an l-dimensional space, it can be expanded in terms of the root vectors, M = p k α k (7) k= Inserting this expansion into eq (6) and using eq (7) yields n j = A jk p k (8) k= Inverting this result gives: p k = (A ) kj n j (9) j= 6
IV A general formula for c R and the second-order index I (R) In a representation R, C (R) M = c R M To compute c R, we employ eq (5) to obtain: C (R) M = (M,M) M + In terms of δ α + (E α E α +E α E α ) M = (M,M) M + α + [E α, E α ] M = (M,M) M + α + (α,m) M α + α, which is defined in eq (), we can write: C (R) M = (M, M +δ) M (30) Using eq (7), ( ) (M, M +δ) M ) = p k α k, M +δ k= = k= p k [(α k, α k )(n k +)] after making use of eq (3) Finally, inserting eq (9) for p k, c R = (α k, α k )(n k +)(A ) kj n j (3) j= k= Eq (3) is our basic result, which has also been obtained in ref [0] This is sometimes rewritten in terms of the symmetrized Cartan matrix, which is defined by [7]: G ij (α j, α j ) A ij = 4(α i, α j ) (α i, α i )(α j, α j ) = (α i, α j ) (3) The inverse of the symmetrized Cartan matrix, which we shall denote by G ij is therefore given by: G ij = (α i, α i )A ij (33) 7
One can immediately check that G ij G jk = δi k as required Hence, eq (3) can be rewritten as []: c R = (n k +)G kj n j (34) j= k= For any irreducible representation, Tr(R a R b ) = I (R)g ab, (35) where I (R) is called the second-order index of the representation R By virtue of eq (4), the second-order index of the adjoint representation is I (A) = For an arbitrary irreducible representation R, taking the trace of eq (6) yields: c R = I (R)d G d R (36) where d G is the dimension of the Lie algebra (which is also equal to the number of generators) and d R is the dimension of the representation For the adjoint representation (R = A), we have d R = d G, in which case we obtain the expected result, c A = I (A) = (37) V The quadratic Casimir operator and second-order index for irreducible representations of su(n), so(n) and sp(n) We begin by listing the inverse Cartan matrices for su(l+), so(l), so(l+) and sp(l), where l is the rank of the corresponding Lie algebras [4] su(l+) (l ) : A = l+ l l l l 3 3 l (l ) (l ) (l 3) 6 4 l (l ) (l ) 3(l 3) 9 6 3 l 3 (l 3) 3(l 3) 4(l 3) 8 4, 3 6 9 3(l ) (l ) l 4 6 8 (l ) (l ) l 3 4 l l l 8
so(l+) (l 4) : A = 3 3 3 3 3 3 4 4 4 4 3 4 l l (l ) 3 4 l l (l ) 3 4 l l l, so(l) (l 4) : A = 3 3 3 3 3 3 4 4 3 4 l (l ) (l ) 3 (l ) 4 l 4 (l ) 3 (l ) 4 (l ) 4 l, sp(l) (l 4) : A = 3 3 3 3 3 3 4 4 4 4 3 4 l l l 3 4 l l l 3 (l ) (l ) l 9
For the cases of l = and l = 3, we have: ( ) ( ) 0 sp() : A =, so(4) : A =, so(5) : A = sp(3) : A =, so(6) : A = 3 0 3 4 4 4 3 4 ( ), so(7) : A = We also need the length of each simple root In the Cartan-Weyl basis introduced above, the length of each simple root is fixed according to eq (8) These can be evaluated explicitly, and the final results are given by [4]: su(l+) : (α k, α k ) =, k =,,,l, (38) l+ l, for k =,,,l, so(l+) : (α k, α k ) = (39) (l ), for k = l, so(l) : (α k, α k ) = sp(l) : (α k, α k ) =, for k =,,,l, (l ), (40) (l ) (l+), for k =,,,l, (4), for k = l, l+ Finally, we need to identify specific irreducible representations of su(n), so(n) and sp(n) We define the fundamental (or defining) representation of the corresponding groups to be the n-dimensional matrix representation that defines the groups SU(n) and SO(n), respectively, and the n-dimensional representation that defines the group Sp(n) In terms of the Dynkin labels, n (n,n,,n l ), the fundamental representations are given by: n = (,0,0,,0), forsu(l+)(forl ),so(l+)(forl ),so(l)(forl 3), andsp(l)(forl ) For the case of so(3), n = for the fundamental three-dimensional representation The reader is warned that what we call Sp(n) is often called Sp(n) in the literature, 3 0
(since n = is the two-dimensional spinor representation) For the case of so(4), n = (, ) for the fundamental four-dimensional representation (since n = (, 0) and n = (0, ) are two inequivalent two-dimensional spinor representations) Eq (3) then yields: su(l+) : c F = (l+) [ (A ) + ] (A ) k [ ] l so(l+) : c F = (A ) + (A ) k +(A ) l (l ) so(l) : c F = sp(l) : c F = [ (A ) + 4(l ) [ (A ) + 4(l+) The above results can be rewritten as: k= k= ] (A ) k k= ] l (A ) k +4(A ) l k= = l(l+) (l+), l, = l l, l, = l 4(l ), l 3, = l+ 4(l+), l su(n) : c F = n n, (n 3), (4) so(n) : c F = n, (n ) (n 5), (43) sp(n) : c F = n+, 4(n+) (n 5) (44) Wenotethatthedimensionsofthefundamentalrepresentations(d F )andtheadjoint representations (d G ) [the latter is equal to the number of generators] of the simple classical Lie algebras are given by: su(n) : d F = n, d G = n, (45) so(n) : d F = n, d G = n(n ), (46) sp(n) : d F = n, d G = n(n+) (47) Using eq (36), one obtains the second-order index of the fundamental representation: su(n) : I (F) =, n (n ), (48) so(n) : I (F) =, n (n 5), (49) sp(n) : I (F) =, (n+) (n ) (50)
We now examine the adjoint representation and check that Theorem is satisfied The Dynkin labels of the adjoint representation are given by: su(n) : n = (, 0, 0,, 0, 0, ), (n 3), (5) so(n) : n = (0,, 0, 0,, 0, 0), (n 5) (5) sp(n) : n = (, 0, 0, 0,, 0, 0), (n ) (53) For su(), the adjoint representation is given by n = For so(n), the adjoint representation corresponds to the antisymmetric part of the Kronecker product of n n However, the cases of n 6 must be treated separately, as n = (0,) is a spinor representation of so(4) and of so(5), whereas n = (0,,0) is a spinor representation of so(6) For so(3), the fundamental and adjoint representations coincide and correspond to n = For so(4), which is semi-simple, the adjoint representation is not irreducible For so(5), the adjoint representation is given by n = (0,) For so(6), the adjoint representation is given by n = (0,,) We now evaluate the quadratic Casimir operator using eq (3) su(l+) : c A = (l+) [ =, l, (A ) +(A ) l +(A ) l +(A ) ll + [ ] l so(l+) : c A = (A ) + (A ) k +(A ) l =, l 3 (l ) so(l) : c A = [ (A ) + 4(l ) k= ] (A ) k =, l 4 [ ] sp(l) : c A = l (A ) + (A ) k +(A ) l =, l l+ k= I have also checked that the cases of su(), so(3), so(5) and so(6) yield c A = k= ] [(A ) k +(A ) kl ] k= VI The dual Coxeter number We now introduce the maximal weight of the adjoint representation, denoted by θ, which also coincides with the maximal positive root One of the basic theorems of Lie algebras states that for a simple Lie algebra, there are at most two roots of In general, for n odd there is one fundamental irreducible spinor representation of so(n) given by n = (0, 0,, 0, ) For n even there are two fundamental irreducible spinor representations of so(n) given by n = (0, 0,, 0,, 0) and n = (0, 0,, 0, 0, )
different length, called long roots and short roots As θ is the maximal root, it must be a long root We can expand θ in terms of the simple roots θ = a k α k (54) k= The Coxeter number of a simple Lie algebra is defined as [7]: h + a k Likewise, we can expand θ θ/(θ, θ) in terms of the dual roots, k= θ = a kα k, k= where a k = (α k,α k ) (θ, θ) a k, after using eqs (4) and (54) The dual Coxeter number is then defined as [7]: g + k= a k = + (θ, θ) (α k, α k )a k (55) For a simply-laced Lie algebra (defined as a simple Lie algebra whose non-zero roots are all of equal length), we have g = h The Dynkin labels for θ, n θ j (θ, α j) (α j, α j ), are given explicitly in eqs (5) (53) for su(n), so(n) and sp(n), respectively Following eqs (8) and (9), we can write k= n θ k = A kj a j, a k = j= (A ) kj n θ j (56) j= It follows that: (θ, θ) = a j a k (α k, α j ) = a j a k (α k, α k )A kj j= k= = k= a k (α k, α k )n θ k = 3 j= k= j= k= (α k, α k )n θ k(a ) kj n θ j
Using eqs (55) and (56), the dual Coxeter number can be rewritten as: g = + (θ, θ) j= k= (α k, α k )(A ) kj n θ j Since θ is the maximal weight of the adjoint representation, it follows from eq (3) that c A = (α k, α k )(n θ k +)(A ) kj n θ j j= k= = (θ, θ)+(g )(θ, θ), which reduces to Since c A =, we conclude that: g = (θ, θ) = c A = g(θ, θ) n, for su(n), (n ),, for so(3), n, for so(n), (n 4), n+, for sp(n), (n ), (57) after using eqs (38) (4) for the length of the long root The second-order index and the eigenvalue of the Casimir operator of the fundamental representation are related very simply to the dual Coxeter number Eqs(48) (50) yield: and eq (36) then gives: su(n) : I (F) =, (n ), (58) g so(n) : I (F) =, (n 5), (59) g sp(n) : I (F) =, (n ), (60) g C F = g GI (F) d F For completeness, we provide an explicit computation of (θ, θ) Multiplying eq () by g ij and summing over i and j yields, (α, α) = l, (6) α 4
where l is the rank of the group This result can be used to compute (θ, θ) as follows All non-zero roots of a simply-laced Lie algebra are of equal length By definition, the maximal root θ is regarded as a long root Since there are d G l non-zero roots, it follows from eq (6) that l = (d G l)(θ, θ), or (θ, θ) = l, for g = su(l+) [l ] and so(l) [l ] d G l For so(l+) [l ], there are l long roots and one short root We use Weyl reflections togeneratetheremainingnon-zeroroots, which resultsin(l )(d G l)/l long roots and (d G l)/l short roots For sp(l), there is one long root and l short roots We use Weyl reflections to generate the remaining non-zero roots, which results in(d G l)/l long rootsand(l )(d G l)/l short roots Hence, for so(l+) [l ], eq (6) yields: [ (l )(dg l) l = l + d ] G l l (θ, θ) = (l )(d G l) (θ, θ), l and for sp(l), eq (6) yields: [ dg l l = + (l )(d ] G l) (θ, θ) = (l+)(d G l) (θ, θ) l l l Therefore, (θ, θ) = l d G l, for g = so(l+), (l ), l, for g = sp(l), (l ) l+ Using eqs (45) (47), we end up with:, n for su(n), (n ), (θ, θ) =, for so(3),, for so(n), (n 4), n, for sp(n), (n ), n+ in agreement with eq (57) Additional properties of the Coxeter number and the dual Coxeter number can be found in ref [3] 5
VII The Strange Formula For completeness, I shall record a formula first obtained by Freudenthal and de Vries [4] This is a formula for the length of the Weyl vector [cf eq ()], (δ,δ) = 4 d G, (6) where d G is the dimension of the Lie algebra This remarkable formula can be verified explicitly for all the simple Lie algebras Elementary proofs of eq (6), known as the strange formula, can be found in refs [5,6] VIII An alternative normalization convention We highlight two implicit normalization conditions employed in this note First, g ab = Tr(F a F b ) defines the Killing metric, which is normalized by a coefficient of Second, the roots are normalized by α i α j = g ij α It is convenient to alter these conventions as follows First, we redefine [7, ] g ab = gη Tr(F af b ), (63) where g is the dual Coxeter number and η is an additional rescaling factor In order to be consistent with eq (), we shall simultaneously rescale the roots so that gη Multiplying by g ij and summing over i and j yields: (α, α) = gηl, α α i α j = g ij (64) α which replaces eq (6) and fixes the length of the root vectors We can identify: η = (θ, θ), since gη = g(θ, θ) = returns us to our previous conventions [cf eq (57)] This rescaling can be viewed in two equivalent ways As presented above, it can be viewed simply as a rescaling of the definition of the Killing metric Note that in this interpretation, the eigenvalue of the Casimir operator and the secondorder index are independent of the choice of basis for the generators, since the 6
definitions given by eqs(5) and(35) are covariant with respect to their indices That is, rescaling the Lie algebra generators automatically rescales the Killing metric, leaving the eigenvalue of the Casimir operator and the second-order index invariant However, we can also view eq (63) as a rescaling of the definition of the Lie algebra generators, with g ab held fixed In practice, one typically chooses the basis for the Lie algebra generators such that g ab = δ ab, where the coefficient in front of the Kronecker delta is held fixed at In this interpretation, the eigenvalue of the Casimir operator and the second-order index depend on the normalization of the Lie algebra generators 3 Of course, both interpretations are equally valid The symmetrized Cartan matrix defined in eq (3) also depends on the overall scale of the roots However, we shall simply redefine it as: G ij = (θ, θ) (α j, α j ) A ij (65) Note that eq (65) is independent of the convention for the normalization of the length of the roots The inverse of the redefined symmetrized Cartan matrix is given by G ij = (α i, α i ) (θ, θ) A ij This matrix is called the quadratic form matrix in ref [7] The explicit forms of the G ij for the simple Lie groups are given in refs [3,7] As noted above, the eigenvalue of the quadratic Casimir operator and the secondorder index are rescaled by ηg, and we shall denote the corresponding rescaled quantities by capital letters, In particular, eq (37) implies that: C R ηgc R, T R ηgi (R) (66) C A = T A = ηg (67) If we multiply eq (34) by ηg, and rescale G as indicated above, we obtain []: C R = ηgc R = η j= k= (a i +)G ij a j, (68) where eq (57) has been used to convert to the new definition of G Finally, wenotethatthestrangeformulaoffreudenthalanddevries[cfeq(6)] is also rescaled by ηg, (δ,δ) = 4 ηgd G (69) 3 In ref [0], this viewpoint is described on the top of p 30 as follows If all generators in a given simple Lie algebra are multiplied with a common factor λ, the structure constants f c ab are multiplied with λ and the Killing form is multiplied with λ For convenience, the inner product in the root space [cf eq ()] is redefined to be Euclidean again, namely, the metric tensor in the root space is δ ij instead of g ij 7
It is often convenient to choose the squared-length of the longest root to be equal to That is, 4 η (θ, θ) = (70) In this convention, our original definition of the symmetrized Cartan matrix defined in eq (3) and the rescaled version defined in eq (65) are of the same form Consequently, when η =, the form of eqs (34) and (68) coincide since both c R and G ij scale in the same way As a consequence of eqs (36), (66) and (67), C F = T Fd G d F, C A = T A = T F I (F) It then follows that: su(n) : C F = T F ( n n ), C A = T A = nt F, (n ), (7) so(n) : C F = T F(n ), C A = T A = T F (n ), (n 5) (7) sp(n) : C F = T F(n+), C A = T A = T F (n+), (n ) (73) Comparing the above results with eqs (57) and (67), it follows that the normalization of the Lie algebra generators are fixed according to []: su(n) : T F = η, (n ), so(n) : T F = η, (n 5), sp(n) : T F = η, (n ) Of course, the above results are consistent with eqs (58) (60), in light of eq (66) As noted in eq (70) and in footnote 4, η = is the common choice in the mathematics literature In contrast, η = is more typically employed in the physics literature, especially in the case of the su(n) Lie algebra Although a universal choice for η is desirable, it is not required As a result, it is not uncommon to see different conventions for η applied to different simple Lie algebras [] For example, the results of eqs (7) (73) agree with Table 3 of ref [5], where T F = has been taken for all su(n), so(n) and sp(n) generators This choice requires a different choice of η for so(n) as compared to su(n) and sp(n) It is also common for physicists to choose T F = for su(n) and sp(n) and T F = for so(n), which again requires a different choice of η for so(n) as compared to su(n) and sp(n) 4 This convention is common in the mathematics literature It is motivated by the observation that in this convention, I (R) is always an integer 8
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