Uniqueness of universal dimensions and configurations of points and lines
aa r X i v : . [ m a t h . QA ] J a n Uniqueness of universal dimensions andconfigurations of points and lines
M.Y. Avetisyan and R.L. Mkrtchyan
Yerevan Physics Institute, Yerevan, Armenia
Abstract
The problem of uniqueness of universal formulae for (quantum) dimen-sions of simple Lie algebras is investigated. We present generic functions,which multiplied by a universal (quantum) dimension formula, preserveboth its structure and its values at the points from Vogel’s table. Con-nection of some of these functions with geometrical configurations, suchas the famous Pappus-Brianchon-Pascal (9 ) configuration of points andlines, is established. Particularly, the appropriate realizable configuration(144 ) (yet to be found) will provide a symmetric non-uniquenessfactor for any universal dimension formula. Introduction
Universal dimension formulae [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] are functions f ( α, β, γ ) of three homogeneous so-called universal, or Vogel’s parameters ( α : β : γ ) - the projective coordinates in Vogel’s plane . These universal parametersparticularly parametrize all simple Lie algebras, by setting up a correspondencebetween specific projective coordinates and the latters. This correspondence canis presented in Vogel’s Table 1. Universal formulae possess a specific structure:they are rational functions, where both the numerator and denominator decom-pose into products of the same number of linear factors of universal parameters: F = k Y i =1 n i α + x i β + y i γm i α + z i β + t i γ (1)The general form of universal quantum dimensions is similar, with the linearfunctions substituted by their hyperbolic sines : In [7] Vogel’s plane is defined as the factor of projective plane by permutations of thehomogeneous parameters. Here we refer as
Vogel’s the projective plane itself. The following notation is used ( x ) = sinh[ x : k Y i =1 n i α + x i β + y i γm i α + z i β + t i γ (2)The characteristic property of a universal (quantum) dimension function isthat at the points in the Vogel’s projective plane, corresponding to each of theLie algebras, it yields (quantum) dimension of some irreducible representationof the particular algebra, extended by the automorphism group of its Dynkindiagram [3, 5]. We also require that at the points, obtained by permutationof the universal coordinates for the initial ones, the output of the universalfunction is equal to a (quantum ) dimension of some irreducible representationof the corresponding simple Lie algebra as well. It is permissible, that at some ofthe abovementioned points a universal function may be singular (see, however,[11, 12], where the linear resolvability feature, which allows to assign relevantvalues to universal dimension formulae at their singular points, is described).As an example, we present the simplest universal quantum dimension, thatis for the adjoint representation ad : f ( x ) = − sinh (cid:16) γ +2 β +2 α x (cid:17) sinh (cid:0) α x (cid:1) sinh (cid:16) γ + β +2 α x (cid:17) sinh (cid:16) β x (cid:17) sinh (cid:16) γ +2 β + α x (cid:17) sinh (cid:0) γ x (cid:1) (3)We also present the universal quantum dimension X ( x, k, n, α, β, γ ) of theCartan products of the k -th power of X and the n -th power of the adjointrepresentations [11]: X ( x, k, n, α, β, γ ) =sinh h x k − Y i =0 ( α ( i − − β ) ( α ( i − − γ ) ( β + γ + α ( − ( i − ( α ( i + 1)) ( β − α ( i − ( γ − α ( i − ×× n Y i =0 ( α ( i + k − − β )( α ( i + k − − γ )( β + γ + α ( − ( i + k − α ( i + k + 1))( β − α ( i + k − γ − α ( i + k − ×× k + n Y i =1 ( − β − γ + α ( i − − β − γ + α ( i − α ( i − − β + γ ))( α ( i − − β )( α ( i − − γ )( β + γ − α ( i − ×× ( α + β )( α + γ )( α ( n + 1))(2 α + 2 β )(2 α + 2 γ )(2 α + β + γ ) ×× ( α (3 k + n − − β + γ ))( α (3 k + 2 n − − β + γ ))(3 α + 2 β + 2 γ )(4 α + 2 β + 2 γ ) (4) sinh [ x : A · B...M · N... ≡ sinh( xA ) sinh( xB ) ... sinh( xM ) sinh( xN ) ... X is the representation, appearing in the following universal decom-position ∧ ad = ad ⊕ X (5)which holds for all the simple Lie algebras.As we see, universal formulae may be sufficiently intricate. One may askthen, whether it is possible to rewrite them more compactly, without violatingtheir structure and preserving their values at the points from Vogel’s table.The investigation, presented in this paper, is a step forward to the answer tothis question. We examine the problem of uniqueness of the universal dimension(quantum dimension) formulae.Let F and F are universal (quantum) dimension formulae, yielding thesame outputs at the points from Vogel’s table, as well as at the correspondingpoints with permuted coordinates. Then their ratio Q - the non-uniquenessfactor is obviously equal to 1 at those points. Q = F F (6)Evidently, Q has the same structure as (1) or (2).So, the question on uniqueness rephrases into the investigation of existenceof such a Q , which is equal to 1 both at the points from Vogel’s table 1, as wellas at all those, obtained by permutations of the associated coordinates.The complete solution of this problem has not been achieved yet, however,we present solutions for some variations of it, as well as present a geometricinterpretation both of the general problem and its variations. This interpreta-tion sets up a remarkable connection between the problem of uniqueness andthe classical study of configurations of points and lines.First, note that the points from Vogel’s table occupy the following distin-guished lines: sl : α + β = 0 , (7) so : 2 α + β = 0 , (8) sp : α + 2 β = 0 , (9) exc : γ − α + β ) = 0 , (10)on which the linear, orthogonal, symplectic and the exceptional algebras aresituated, respectively.We add an additional condition to the initial problem, namely, we requirethat Q is equal to 1 not only at the points, associated with the simple Liealgebras, but also on the entire distinguished lines. In case of the sl, so, sp lines this requirement holds anyway, since a rational function, yielding 1 at theinteger-valued coordinates on the halfline, is always equivalent to 1 on the entire3ine. However, in case of the exc line this requirement is a non-trivial additionalone, since it is a variation of Deligne’s hypothesis [3, 4], which has not beenproved yet.Next, the initial lines, as well as those, obtained by permutations (i.e. lines,obtained by permutations of the coordinates) collect into a set of 12 lines, onwhich Q is equivalent to 1; it includes the abovementioned sl, so and exc lines(we shall call them basic lines), and the following lines, obtained by all permu-tations, among which there is the sp line as well (we omit = 0 part in the lines’equations below): α + γ, β + γsp : α + 2 β, α + 2 γ, β + 2 γ, γ + 2 α, γ + 2 β (11) β − α + γ ) , α − β + γ ) , Our investigation is built in the following way: first, we require that Q ≡ sl, so, exc , and derive a general non-uniquenessfactor for this case in Section 1.1.We then impose the requirement for Q ≡ sp line - one of the lines,obtained by the so line after the α ↔ β permutation of the parameters, and findseveral solutions, employing some results taken from the theory of configurationsof points and lines in Section 2.The further expansion of the Q ≡ Q for universal dimensions To simplify calculations, we make the following change of coordinates : α ′ = α + β α = − α ′ + β ′ β ′ = 2 α + β β = 2 α ′ − β ′ γ ′ = γ − α + β ) γ = 2 α ′ + γ ′ such that in the primed coordinates the equations of the basic lines sl, so, exc will simply be α ′ = 0 , β ′ = 0 , γ ′ = 0 . The line equations (11) in terms of these new coordinates rewrite as: α ′ + β ′ + γ ′ , α ′ − β ′ + γ ′ α ′ − β ′ , α ′ + β ′ + 2 γ ′ , α ′ − β ′ + 2 γ ′ , β ′ + γ ′ , α ′ − β ′ + γ ′ (12) − α ′ + 3 β ′ − γ ′ , − β ′ − γ ′ From now on we work with the primed parameters, dropping the prime markfor convenience (up to special notice). 4 .1 Q for universal dimensions: sl, so, exc and sp lines Let’s take the general form (1) of Q , assuming it is written in terms of the primedparameters, and consider its values on the three lines α = 0 , β = 0 , γ = 0 . We require Q ≡ α = 0. Then1 ≡ k Y i =1 x i β + y i γz i β + t i γ (13)and one deduces, that z i = l i x q ( i ) , t i = l i y q ( i ) , with some permutation q ( i ) , i = 1 , ...k , and non-zero multipliers l i with l l ...l k = 1.Substituting these relations into (1), one has Q = k Y i =1 n i α + x i β + y i γm i α + l i x q ( i ) β + l i y q ( i ) γ (14)Absorbing the 1 /l i into m i , renumbering m i → m q ( i ) and changing the orderof the multipliers in the denominator, we rewrite Q as: Q = k Y i =1 n i α + x i β + y i γm i α + x i β + y i γ (15)Now let Q ≡ β = 0: 1 ≡ k Y i =1 n i α + y i γm i α + y i γ (16)Then one must have y i = k i y s ( i ) , m i = k i n s ( i ) , with some permutation s ( i )and with k k ...k k = 1, so that Q accepts the form: Q = k Y i =1 n i α + x i β + y i γk i n s ( i ) α + x i β + k i y s ( i ) γ = k Y i =1 n i α + x i β + y i γk i n s ( i ) α + x i β + y i γ (17)Next we require Q ≡ γ = 0:1 ≡ k Y i =1 n i α + x i βk i n s ( i ) α + x i β (18)Again, from this relation we infer x i = c i x p ( i ) (19) k i n s ( i ) = c i n p ( i ) (20)5or some permutation p ( i ) and c i with c c ...c k = 1.So, altogether we have the following expression for Q with the restrictionson its parameters: Q = k Y i =1 n i α + x i β + y i γk i n s ( i ) α + x i β + y i γ = k Y i =1 n i α + x i β + y i γc i n p ( i ) α + x i β + y i γ (21) x i = c i x p ( i ) (22) y i = k i y s ( i ) (23) k i n s ( i ) = c i n p ( i ) (24) c c ...c k = 1 (25) k k ...k k = 1 (26)for some permutations s ( i ) , p ( i ). Note that after having solved these equations,one must check the Q on absence of any cancellation in it.It is easy to show, that there is not a non-trivial solution if k = 1 ,
2. For k = 3 one can show that the existence of a non-trivial solution requires thatthe permutations s ( i ) , p ( i ) do not have fixed points and do not coincide, i.e. s ( i ) = i + 1 , p ( i ) = i + 2 ( mod n i = 0,so that one can factor them out, or effectively put n i = 1, so that k i = c i , y = c y , y = c c y , x = c x , x = c c x . Denoting x = x, y = y , we getthe final expression of Q :( α + βx + γy )( αc c + βc x + γy )( αc + βc c x + γy )( αc + βx + γy )( α + βc x + γy )( αc c + βc c x + γy ) (27)Finally, we require Q ≡ α − β = 0:1 ≡ k Y i =1 α ( n i + 3 x i ) + y i γα ( c i n p ( i ) + 3 c i x i ) + y i γ (28)which leads to c i n p ( i ) + 3 x i = r i ( n v ( i ) + 3 x v ( i ) ) (29) y i = r i y v ( i ) (30) k Y i =1 r i = 1 (31) i = 1 , , ..., k (32)for some permutation v ( i ).So, altogether we have the expression of Q with the following restrictions onits parameters: 6 = k Y i =1 n i α + x i β + y i γk i n s ( i ) α + x i β + y i γ = k Y i =1 n i α + x i β + y i γc i n p ( i ) α + x i β + y i γ (33) x i = c i x p ( i ) (34) y i = k i y s ( i ) (35) k i n s ( i ) = c i n p ( i ) (36) y i = r i y v ( i ) (37) c i n p ( i ) + 3 x i = r i ( n v ( i ) + 3 x v ( i ) ) (38) c c ...c k = 1 (39) k k ...k k = 1 (40) r r ...r k = 1 (41)for some permutations s ( i ) , p ( i ) , v ( i ) , i = 1 , ...k .Evidently, one can keep on going in this manner line after line, requiring Q ≡ k i , satisfying k k ...k k = 1. Then, provided these introduced coefficients aregiven, one has to solve the linear equations on the parameters. Finally, oneshould check whether the solution is non-trivial, i.e. make sure, that there is nocancellation in Q . Since the number of permutations is k !, a lot of candidatesfor the solution will be obtained at each of the steps (each line).We shall not try to solve these equations directly, instead, we will derive asolution for k = 4, after setting up a geometrical interpretation of the problemand using several results from the theory of configurations of points and linesin Section 2. Q for universal quantum dimensions Lemma : If sinh[ x : k Y i =1 n i m i ≡ then the { n i , i = 1 , ..., k } and { ǫ i m i , i = 1 , ..., k } sets coincide for somechoice of ǫ i = ± , i = 1 , ...k , Q ki =1 ǫ i = 1 .Proof. Without loss of generality, let | n k | = max ( {| n | , ..., | n k | , | m | , ..., | m k |} ).Then some of the factors in the numerator of Q is zero at x = iπ/n k . To cancelthis zero out, there is necessarily a multiplier in the denominator, zeroing at thesame point. Thus iπ/n k = iπq/m i , for some q ∈ Z , from which it follows that q = ± m i = ± n k . Applying the same logic for the remaining factors, weprove the statement of the lemma. 7ow consider the general expression (2) of Q : Q ( x ) = sinh[ x : k Y i =1 n i α + x i β + y i γm i α + z i β + t i γ (43)Due to the Lemma the analysis of this expression is almost the same as thatin the previous subsection. The only difference is that the parameters k i , c i , r i ,must now be equal ±
1, so that the general solution for Q is given by the formula Q ( x ) = sinh[ x : k Y i =1 n i α + x i β + y i γk i n s ( i ) α + x i β + y i γ = sinh[ x : k Y i =1 n i α + x i β + y i γc i n p ( i ) α + x i β + y i γ (44)with the same sets of equations: (22) - (26) for the case of three lines - sl, so, exc , and (34) - (41) for the case of four lines sl, so, sp, exc , with additionalrequirements k i = ± , c i = ± , r i = ± , i = 1 , ...k .It is easy to prove, that in case of three lines, i.e. equations (22) - (26)there is not a non-trivial solution Q ( x ) for the quantum dimensions for k ’s upto k = 3, inclusively, so that the first non-trivial solution exists for k = 4.A solution, obtained by employing some knowledge from the theory of con-figurations, is presented below. In this section we provide a geometric point of view to the problem of uniqueness,setting up its connection with a classical problem of the so-called configurations ,namely configurations of points and lines . First, observe, that each of the linear factors in the expression of Q correspondsto a line in the projective Vogel’s plane. Indeed, to each of the factors xα + yβ + zγ one can correspond a line equation: xα + yβ + zγ = 0.Thus, for any given expression for universal (quantum) dimension, with say k multipliers, we can draw a unique picture in the Vogel’s plane, consisting of k lines, corresponding to the linear factors in numerator, which will be referredas red lines for convenience, and k green lines for those in the denominator. Inaddition, we can draw a number of black lines, corresponding to the distinguished sl, so, exc , lines as well as those, associated to the permuted coordinates - suchas the sp line. 8ne can see the corresponding picture for the simplest universal formula,namely the dimension of the adjoint representation (3), in Figure 1.Let’s consider the picture, associated to a non-uniqueness factor Q . It turnsout that each of the black lines must contain k points, at which a green and ared line intersect.Indeed, this statement exactly rephrases the cancellation mechanism, de-scribed in the previous section: when restricting Q to a black line, each of thefactors from the numerator is proportional to some factor from the denomina-tor. This means that these two factors are zeroing simultaneously, meaning,that residing on a black and, say, a red line at once, we necessarily reside on agreen line too.It is easy to notice, that this corresponding picture also contains informationabout the choice of the permutations s ( i ) , p ( i ) , ... , (see equations (33)-(41)), -the intersection points of three different-colored lines obviously define the pairsof cancelling factors, when restricting the function to each of the distinguishedblack lines.Thus, the picture of k black, k red and k green lines, corresponding to anon-uniqueness factor Q , has the following characteristic feature: on each of theblack lines there are k points at which a red and a green line intersect. Notethat besides these points of intersection of three differently colored lines, theremay be some other intersection points, which however will not be of interest forus. Let’s introduce the following standard definitions [13, 14]:
Definition 1.
We say a line is incident with a point, (equivalently, a point is incident witha line) if it passes through it (equivalently, if it lies on it).
Definition 2. A configuration ( p γ l π ) is a set of p points and l lines, such that everypoint is incident with precisely γ of these lines and every line is incident withprecisely π of these points. Remark 1.
Notice, that the total number of incidences, on one hand, isequal to pγ , and is lπ , on the other hand, so that from Definition 2 it follows,that pγ = lπ . Remark 2. If p = l, γ = π , the configuration is denoted by ( p γ ).We see that the picture of k black, k red and k green lines, possessing thefeature described in the previous subsection, turns into a configuration iff thenumber of black, red and green lines coincide and is equal to k . Obviously, thecorresponding configuration will be ( k , k k ).However, if we have a configuration ( k , k k ), it doesn’t mean that we candefinitely construct a corresponding Q . The possible obstacle is that one would The labeling of the lines in the following figures is meant to identify the correspondingcolors they are given. For example, in Figure 1, r identifies the line, associated to the secondfactor in the numerator of (3), and g - to the third factor in the corresponding denominator. k lines such thatat each of the points, belonging to the configuration, three lines of differentcolors meet. Such configuration are presented in Figures 3 and 4.For any given configuration ( p γ l π ) one can construct a so-called configurationtable : we label the points and lines of that configuration, then for each of thelines allocate a column, consisting of the labels of the points, which are incidentwith the corresponding line. Characteristic properties of a configuration tableare the following: the label for each of the points occurs in exactly γ columns,different columns do not contain two similar labels of points, and each columncontains exactly π labels. Two configuration tables are identical, if they coincideafter some relabeling of points and lines, and/or rearranging the points in a givencolumn.So, ”possible” configurations of a given type ( p γ l π ) can be considered simplyas different configuration tables of that type.Further, a configuration table is called realizable , if one can construct ageometrical picture of lines and points corresponding to it. Not all tables arerealizable. (9 ) configuration and Q for the sl, so, exc lines A relevant configuration happens to be corresponding to the solution (27), de-rived for k = 3, which is equivalent to 1 on three basic lines - (8), (9), and (10).It is the configuration (9 , ) , which is usually referred as (9 ) [13, 14], sincethe terms in the standard notation coincide. This configuration is also known asthe Pappus (Pappus of Alexandria) or Pappus-Brianchon-Pascal configuration,which is presented in Figure 2.The index in the notation (9 ) is to indicate the fact, that there are several(9 ) configurations, so that it is used to distinguish these. Possible values of theindex, i.e. the number of different configurations (9 ) is 3, equivalently, thereare three different configuration tables for (9 ) configuration. Each of these 3tables happens to be realizable. However, only one of them, presented in Figure2, (9 ) from [13], can be colored in the way we need. For example, for theconfiguration (9 ) (see Figure 3) it is impossible to distinguish 3 black lines,since for any two lines of the configuration, there is always a third one, whichintersects with one of them at some point, which belongs to the configuration.This violates the requirement, that at each point of the configuration three linesof different colors intersect. The same reasoning holds for the remaining thirdconfiguration (9 ) - the (9 ) , see Figure 4.Obviously, the solution (27) for k = 3 is in one-to-one correspondence withthe Pappus-Brianchon-Pascal (9 ) configuration, since there is no other (9 )configuration which can be colored in a needed way.The picture, associated to the non-uniqueness factor (27) is given in Figure5. The latter is the configuration (9 ) after a projective transformation, whichtakes the α = 0 line to infinity. One has to take into account that as the equations of the three distinguished lines are c , c , x, y in the (27)expression of Q are easily observed in the same Figure 5, where the associatedcoordinates of the points of the configuration are shown explicitly. (16 ) configuration and Q for the sl, so, exc and sp lines Let’s now take 4 black lines - sl, so, sp, exc , and search for a non-uniquenessfactor Q , which is equivalent to 1 on those. If we take k = 4, we happen to bedealing with the configuration (16 ). One of its realizations, taken from [14],presented in Figure 2.It turns out, that it is possible to color the lines of this particular configu-ration as needed: with the chosen coloring, demonstrated in Figure 2, at eachof the points of the configuration three lines of different - black, red, and green- colors meet.After labeling the lines, we track the incidence relations, which right awayidentify the patterns of cancellation of the factors in Q , or, equivalently, thechoice of permutations in (33)-(41) equations; permutations, dictated by thisconfiguration are easily defined and are as follows: s (1) = 2 , s (2) = 1 , s (3) = 4 , s (4) = 3 p (1) = 4 , p (2) = 3 , p (3) = 2 , p (4) = 1 v (1) = 3 , v (2) = 4 , v (3) = 1 , v (4) = 2In fact, this set of permutations is the main (and only) information we takeout of a given configuration. Then, having these permutations at hand, we solvethe equations (34) - (41), to derive the required non-uniqueness factor.Equations (34)-(36), derived for three distinguished lines: α = β = γ = 0,yield x = x c , x = x c k k (45) y = y k , y = y k (46) n = n k c k k , n = n k c (47) k = 1 k , k = 1 k , (48) c = 1 c k k , c = c k k (49)The remaining (37)-(38) equations yield α = 0 , β = 0 and γ = 0, one of them unavoidably will be the ideal line of the projective plane,i.e. the line in the infinity (we choose α = 0). = 1 r , r = r k k , r = k r k (50) y = r y (51)and r = ± c k (52)The case with the plus sign, i.e. when r = c k leads to n = k n , x = k x (53)which produces a trivial Q , i.e. the solution is Q ≡ r = − c k , then: n = − n + 3 x + 3 k x k (54)Overall, one substitutes the values of all variables, given above, expressed interms of the remaining arbitrary variables k , k , c , x , x , y , n into the generalexpression for Q . Note, that in case of quantum dimension the coefficients k , k , c are equal to ±
1. However, one can absorb some variables and define: n = n , y = c k y , x = x , x ′ = k x (55)Then the final expression for a dimension non-uniqueness factor Q is Q = nα + xβ − yγ − ( n + 3 x + 3 x ′ ) α + xβ − yγ · − ( n + 3 x + 3 x ′ ) α + x ′ β − yγnα + x ′ β − yγ × nα + x ′ β + yγ − ( n + 3 x + 3 x ′ ) α + x ′ β + yγ · − ( n + 3 x + 3 x ′ ) α + xβ + yγnα + xβ + yγ (56)The corresponding quantum dimension non-uniqueness factor Q appears towrite almost similarly - just with the sinh [ x : signs inserted. Q ( x ) = sinh[ x : nα + xβ − yγ − ( n + 3 x + 3 x ′ ) α + xβ − yγ · − ( n + 3 x + 3 x ′ ) α + x ′ β − yγnα + x ′ β − yγ × nα + x ′ β + yγ − ( n + 3 x + 3 x ′ ) α + x ′ β + yγ · − ( n + 3 x + 3 x ′ ) α + xβ + yγnα + xβ + yγ (57)12 .5 The (144 ) configuration and a symmetric non-uniquenessfactor Q An immediate problem, arising after the previous investigation, is the derivationof a symmetric non-uniqueness factor Q , which would be equivalent to 1 on allthe 12 lines, obtained by the basic lines after all possible permutations of thecoordinates. The search of such a Q appears to be one of a realizable (144 )configuration in the scope of the geometrical approach. Unfortunately, this con-figuration has not been studied yet, so the existence of a symmetric Q remainsan open question. Another possible approach for searching a totally symmetric Q , which would beequivalent to 1 on all lines (7) - (11), is based just on its symmetry. If we findsuch a Q , which is 1 on all three basic lines sl, so, exc and is symmetric w.r.t.all permutations of the universal parameters, then it would be equivalent to 1on all permuted lines as well. It is also sufficient to require its invariance w.r.t.the generating elements of the S group of permutations, say the transpositions α ↔ β and β ↔ γ (in unprimed variables). So in fact, a possible strategy is totake the general solution (21) for some k (perhaps at least k = 12), and requireits symmetry w.r.t. these two permutations, which in terms of the primedparameters looks like: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ′ β ′ γ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ′ α ′ − β ′ γ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ′ β ′ γ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ′ + β ′ + γ ′ β ′ + γ ′ − β ′ − γ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (58)repectively. Conclusion
We examine the problem of uniqueness of universal (quantum) dimensions ofthe simple Lie algebras. In other words, we search for non-uniqueness factors,i.e. non-trivial expressions, yielding 1 at the points from Vogel’s table. Theyobviously preserve the values of a universal formula at the corresponding point,when multiplying them by the latter.We derive several non-uniqueness factors, particularly one, which is equiv-alent to 1, when restricting it to each of the three - sl, so, exc - distinguishedlines in the Vogel’s plane, as well as another one, which is equivalent to 1 oneach of the four distinguished lines - sl, so, sp, exc .Derivation of the latter has been carried out, using the remarkable connectionof the problem of uniqueness with the theory of configurations of points andlines, which we establish by setting up a geometric interpretation of the non-uniqueness factors; it particularly corresponds to the (16 ) configuration,presented in Figure (6). 13ore interestingly, this geometric interpretation of the uniqueness problemreveals the one-to-one correspondence of the non-uniqueness factor, which yields1 on three distinguished lines, with the famous (9 ) Pappus-Brianchon-Pascalconfiguration, known since the 4th century AD!However, the non-uniqueness factors, presented above, will violate the rea-sonable outputs of universal formulae, when considering it after some permu-tation of the coordinates [10, 11]. To preserve the outputs at the points withpermuted coordinates, one needs to have a completely symmetric factor. In thescope of the theory of configurations, the required non-uniqueness factor willprobably be connected with the (144 ) configuration, which is yet to bestudied.The problem of uniqueness deserves a further investigation. Its connectionwith the classical problem of configurations of points and lines is intriguing andseems to have a potential of possible influence in both directions. Acknowledgments
We are indebted to the referee of our paper [11] for a question which is partiallyanswered by the present investigation. MA is grateful to the organizers of RDPOnline Workshop on Mathematical Physics (December 5-6, 2020) for invitation.The work of MA was fulfilled within the Regional Doctoral Program onTheoretical and Experimental Particle Physics Program sponsored by Volkswa-genStiftung. The work of MA and RM is partially supported by the ScienceCommittee of the Ministry of Science and Education of the Republic of Armeniaunder contract 20AA-1C008.Table 1: Vogel’s parameters for simple Lie algebras and the distinguished linesAlgebra/Parameters α β γ t
Line sl ( N ) -2 2 N N α + β = 0 so ( N ) -2 4 N − N − α + β = 0 sp (2 N ) -2 1 N + 2 N + 1 α + 2 β = 0 exc ( n ) − n + 4 n + 4 3 n + 6 γ = 2( α + β )On the exc line n = − / , , , , , G , so (8) , F , E , E , E , respectively.14 g r r r sp sl so exc Figure 1: The ”sketch” of the dimension formula (3).15 r r g g g sl : α = 0 so : β = 0 exc : γ = 0Figure 2: The Pappus-Brianchon-Pascal, or (9 ) configurationFigure 3: The (9 ) configuration, which cannot be ”colored” in order to becorresponded to some Q ) uncolorable configuration References (2011), no. 6, 1292-1339.[3] P. Deligne, La s´erie exceptionnelle des groupes de Lie, C. R. Acad. Sci.Paris, S´erie I (1996), 321-326.[4] P. Deligne and R. de Man, La s´erie exceptionnelle des groupes de Lie II,C. R. Acad. Sci. Paris, S´erie I (1996), 577-582.[5] A. M.Cohen and R. de Man, Computational evidence for Deligne’s conjec-ture regarding exceptional Lie groups, Comptes Rendus de l’Acad´emie desSciences, S´erie 1, Math´ematique, (1996) 322(5), 427-432[6] B.Westbury, Invariant tensors and diagrams, Proceedings of the TenthOporto Meeting on Geometry, Topology and Physics (2001). Vol. 18. Oc-tober, suppl. 2003, pp. 49-82.[7] J.M. Landsberg and L.Manivel, A universal dimension formula for complexsimple Lie algebras. Adv. Math. (2006), 379-407[8] R.L. Mkrtchyan and A.P.Veselov, Universality in Chern-Simons theory.JHEP08 (2012) 153, arXiv:1203.0766.170 ,
0) ( − c x ,
0) ( − c /x, , − c y )(0 , − y ) ( − /x, , − c c y ) r r r g g g exc : γ = 0 so : β = 0Figure 5: The Pappus-Brianchon-Pascal (9 ) configuration after a projectivetransformation 18 g g g r r r r exc : γ = 0 so : β = 0 sl : α = 0 sp : β = 3 α Figure 6: The 16 configuration199] R.L.Mkrtchyan, On Universal Quantum Dimensions, arXiv:1610.09910,Nuclear Physics B921, 2017, pp. 236-249,[10] M.Y. Avetisyan and R.L. Mkrtchyan, X Series of Universal Quantum Di-mensions, arXiv:1812.07914, J. Phys. A: Math. Theor. Volume 53, Number4, 045202, https://doi.org/10.1088/1751-8121/ab5f4d[11] M.Y. Avetisyan and R.L. Mkrtchyan, On ( ad ) n ( X ) kk