aa r X i v : . [ m a t h . R T ] A p r VIRTUAL REPRESENTATION MOTIVES
LIEVEN LE BRUYN
Abstract.
Principal GL n -bundles (aka vector bundles) are locally trivial inthe Zariski topology, whereas principal P GL n -bundles (aka Azumaya alge-bras) are not, to the delight of every non-commutative algebraist. Still, thismakes the calculation of motives of representation schemes of algebras next toimpossible. In very special cases, Brauer-Severi schemes (and their motives)can be used to tackle this problem inductively. We illustrate this in the caseof certain superpotential algebras. The scheme of n -dimensional representations rep n R of a finitely presented non-commutative algebra R R = C h X , . . . , X m i ( F , . . . , F k )is by definition the zero-set in A mn = M n ( C ) ⊕ m of kn equations F l ( A , . . . , A m ) ij = 0 1 ≤ l ≤ k, ≤ i, j ≤ n where A u is the generic n × n matrix, that is, the n × n matrix with entries thecoordinates in A mn = M n ( C ) ⊕ m corresponding to the u -th factor. Very little isknown about the global structure of these representation schemes. Here we will beinterested in their (virtual) motives.1. Motives
Motives are best thought of as a Lego-version of varieties. That is, for everyreduced C -variety X we have, up to Zariski isomorphism, one block [ X ], called the motive of X , which are elements in the ring of (naive) motives Mot C , in whichaddition and multiplication are subject to the following rules: • if Y is a Zariski closed subvariety of X , then we have a ’scissor-relation’[ X ] = [ X − Y ] + [ Y ]allowing us to slice up a variety is locally closed parts and compute itsmotive by adding up these smaller blocks. • if X is a fiber bundle in the Zariski topology with base Y and fiber F , thenwe have a factoring-relation[ X ] = [ Y ] × [ F ]giving us in particular that the motive of a product is the product of themotives. The
Lefschetz-motive L = [ A C ] is the motive of the affine line, and all varietiesallowing a cell-decomposition (such as projective spaces or Grassmannians) havetherefore as their motive a polynomial in L . It is quite easy to verify that[ P n ] = L n +1 − L − , [ GL n ] = n − Y k =0 ( L n − L k ) , [ Gr ( k, n )] = [ GL n ][ GL k ][ GL n − k ] L k ( n − k ) Motives can be calculated either via geometric insight or by laborious algebraicmanipulations. Consider for example a smooth affine quadric in A Q = V ( xy − z − ⊆ A To a geometer this is the affine piece of a smooth quadric in P C which she knows tobe isomorphic to P × P from which she has to remove the intersection with thehyperplane at infinity, which is a P , so its motive must be the difference[ Q ] = ( L + 1)( L + 1) − ( L + 1) = L + L An algebraist would chop up the variety in smaller pieces by localization andeliminating variables. On the open piece x = 0 he can eliminate y = z + 1 x so [ x = 0] = L ( L − z is a free variable and x cannot be zero. On the complement x = 0 theequation becomes z = 1 and therefore its motive is[ x = 0] = 2 L because z = ± y is a free variable. Adding these contribution he gets[ Q ] = [ x = 0] + [ x = 0] = L ( L −
1) + 2 L = L + L Similarly, for a singular affine quadric in C C = V ( xy − z ) ⊆ C our geometer will view this as a cone over a smooth conic in P so would think ofit as P × C ∗ ⊔ { top } with corresponding motive[ C ] = ( L + 1)( L −
1) + 1 = L The algebraist would again decompose into[ x = 0] = [ y = z , x = 0] = L ( L − x = 0] = [ z = 0 , x = 0] = L because y is still a free variable, giving the same answer L . IRTUAL REPRESENTATION MOTIVES 3 ¬ Luna
Let us return to the representation motives [ rep n R ]. An evident approach torepresentation varieties is via invariant theory. That is, consider the action of GL n on rep n R via basechange (conjugation), and study the corresponding quotient-variety π : rep n R ✲✲ rep n R/GL n = iss n .R By Mumford’s GIT the points of iss n R correspond to closed orbits which by a resultof Michael Artin we know are the isomorphism classes of n -dimensional semi-simple representations of R . The quotient map π sends an n -dimensional R -module to theisomorphism class of the direct sum of its Jordan-H¨older components.So, we might try to decompose the quotient variety iss n R according to differentrepresentation types τ of semi-simples and calculate the motives [ iss n R ( τ )] of theseso called Luna strata .In rare cases, for example if the representation scheme rep n R is a smooth variety,one can show that the fibers π − ( S τ ) are isomorphic for all S τ ∈ iss n R ( τ ), so wemight hope in such cases to arrive at the representation motive via[ rep n R ] ? = X τ [ π − ( S τ )][ iss n R ( τ )]Even in the simplest of cases one obtains nonsense.Let R = C [ x ], then clearly, rep C [ x ] = M ( C ) determined by the matrix-imageof x , and the GL -action is by conjugation. The quotient map assigns to a matrixthe coefficients of its characteristic polynomial, so the quotient map is rep C [ x ] = M ( C ) π ✲✲ C = iss C [ x ] A ( T r ( A ) , Det ( A ))Semi-simple matrices are the diagonalisable ones, so there are two representationstypes of semi-simples: τ with two distinct eigenvalues and τ with two equal eigen-values. The second stratum is determined by the closed subvariety V ( Det − T r )which is a smooth conic in C . Therefore,[ iss C [ x ]( τ )] = L and [ iss C [ x ]( τ )] = L − L The fibers π − ( S τ ) are the closed orbits O ( (cid:20) λ λ (cid:21) )which are all a smooth affine quadric in C and therefore [ π − ( S τ )] = L + L .On the other hand, the fibers π − ( S τ ) are the orbit-closures O ( (cid:20) λ λ (cid:21) )which are singular affine quadrics in C , with the top corresponding to the diagonalmatrix. Therefore, [ π − ( S τ ] = L . The Luna stratification approach gives us X τ [ π − ( S ) τ ][ iss C [ x ]( τ )] = ( L + L )( L − L ) + L . L = L + L − L which is clearly different from [ rep C [ x ]] = L . LIEVEN LE BRUYN
What went wrong here is that the fibrations considered are fibrations locallytrivial in the ´etale topology but not in the Zariski topology. Going from coefficientsof the characteristic polynomial to eigenvalues involves taking roots, which aretypical examples of ´etale extensions, but of course not isomorphisms.And we can’t allow ´etale isomorphisms in defining motives because then the ringof motives would become the trivial ring.Another way to explain this difficulty is to observe that the GL n -action on rep n R is actually an action of P GL n and that there is a huge difference between these twogroups when it comes to fibrations. GL n is a special group meaning that all ´etaleprincipal fibrations are in fact Zariski fibrations. Principal GL n -fibrations over anaffine scheme X correspond to rank n projective modules over C [ X ].On the other hand, principal ´etale P GL n -fibrations over X correspond to Azu-maya algebras A over C [ X ], that is, algebras A which are projective modules ofrank n over their center C [ X ] such that A ⊗ C [ X ] A op ≃ End C [ X ] ( A )The correspondence is given by assigning to an Azumaya algebra A over X itsrepresentation scheme. Then, the quotient map rep n A π ✲✲ X = iss n A is the corresponding principal P GL n -fibration.The principal Zariski P GL n -fibrations correspond to the trivial Azumaya alge-bras, that is, those of the form A = End C [ X ] ( P ) where P is a projective C [ X ]-module of rank n .This distinction between ´etale and Zariski principal P GL n -fibrations is at thevery heart of non-commutative algebra. The obstruction to all Azumaya algebrasover X being trivial is measured by an important invariant, the Brauer group Br ( X )of X . 3. Framing
We have seen that we cannot use the Luna stratification approach in order tocompute representation motives, caused by the fact that the acting group on rep-resentation schemes is
P GL n rather than GL n .To bypass this problem we might try to replace the action of P GL n by one of GL n . One way to achieve this is by a process called framing .Instead of rep n R we consider the product rep n R × C n and the action of GL n on it defined by g. ( φ, v ) = ( g.φ.g − , g.v )As long as v = 0 we see that non-trivial central elements act non-trivially on thesecond factor, so this is a genuine GL n -action.So, we might try a stratification strategy on the GL n -variety rep n R × ( C n − { } )in order to compute its motive, which is ( L n − rep n R ], by summing over thedifferent strata.Let us first consider the case when A is an Azumaya algebra. Then, GL n actsfreely on rep n A × ( C n − { } ), and so the corresponding quotient map is a principal IRTUAL REPRESENTATION MOTIVES 5 GL n -fibration π : rep n A × C n − { } ✲✲ ( rep n A × C n − { } ) /GL n = BS n ( A )where BS n ( A ) is called the Brauer-Severi variety of the Azumaya algebra A . Asthis time the quotient map is a Zariski fibration we have[ rep n ( A )]( L n −
1) = [ BS n ( A )][ GL n ]That is, we can compute the representation motive of A if we can compute itsBrauer-Severi motive.In the trivial case when A = M n ( C ) we have that rep n M n ( C ) = P GL n and BS n ( M n ( C )) = P n − so the above equality reduces to[ P GL n ]( L n −
1) = [ P n − ][ GL n ] that is [ P n − ] = L n − L − R the situation is of course more complicated, but we canuse the above idea to compute representation motives inductively from knowledgeof motives of generalised Brauer-Severi varieties.In the product rep n R × ( C n − { } ) let us consider the Zariski open subset of stable couples S n,n ( R ) = { ( φ, v ) | φ ( R ) v = C n } on which GL n acts freely, so we have a principal GL n -fibration S n,n ( R ) ✲✲ S n,n ( R ) /GL n = BS n ( R )with BS n ( R ) the n -the Brauer-Severi variety of R as introduced by Michel Vanden Bergh .We can decompose the product rep n R × ( C n − { } ) into the locally closed strata S n,k ( R ) = { ( φ, v ) | dim C φ ( R ) v = k } giving us this motivic equality( L n − rep n R ] = n X k =1 [ S n,k ( R )] with [ S n,n ( R )] = [ BS n ( R )][ GL n ]In order to calculate the motives of the intermediate srata S n,k ( R ) with 1 ≤ k ≤ n − ψ sending a couple ( φ, v ) to the k -dimensional subspace V = φ ( R ) .v of C n ψ : S n,k ✲✲ Gr ( k, n )To compute the fiber ψ − ( V ) take a basis of V and extend this to a basis for C n ,then with respect to this basis, any couple in the fiber can be written as( φ, v ) = ( (cid:20) φ e φ (cid:21) , (cid:20) w (cid:21) )with ( φ , w ) ∈ S k,k ( R ) and φ ∈ rep n − k ( R ) and e ∈ Ext R ( φ , φ ) an extension ofthe two representations. M. Van den Bergh,
The Brauer-Severi scheme of the trace ring of generic matrices , NATOASI Vol. 233, 333-338 (1987)
LIEVEN LE BRUYN
In extremely rare situations it may happen that this extension-space is of con-stant dimension, say d , along S n,k ( R ), which would then allow us to compute[ S n,k ( R )] = L d [ Gr ( k, n )][ S k,k ( R )][ rep n − k ( R )]= L d [ Gr ( k, n )][ GL k ][ BS k ( R )][ rep n − k ( R )]If we were so lucky for this to hold for all intermediate strata, we would then beable to compute the representation motive [ rep n ( R )] inductively from knowledgeof the representation motives [ rep k ( R )] for k < n and the Brauer-Severi motives[ BS k ( R )] for k ≤ n .Clearly, one would expect this extension condition to hold only for algebras closeto free- or quiver-algebras, and not in more interesting situations. Surprisingly, onecan reduce to the almost free setting in the case of superpotential algebras .4. Superpotentials A superpotential is a non-commutative homogeneous word W ∈ C h X , . . . , X m i d of degree d in m variables. It determines a Chern-Simons functional T r ( W ) : M n ( C ) ⊕ m ✲ C ( M , . . . , M n ) = T r ( W ( M , . . . , M n ))For ringtheorists the relevant fact is that the degeneracy locus of this map { dT r ( W ) = 0 } = rep n R W where R W = C h X , . . . , X m i ( ∂ X , . . . , ∂ X m )is the representation variety of the corresponding Jacobi algebra where the ∂ X i arethe cyclic derivatives with respect to X i .As an example, take W = aXY Z + bXZY + c ( X + Y + Z ), then these cyclicderivatives are ∂ X : aY Z + bZY + cX ∂ Y : aZX + bXZ + cY ∂ Z : aXY + bY X + cZ so the Jacobian algebra R W is the 3-dimensional Sklyanin algebra.The fibers of the Chern-Simons functional M n ( λ ) = T r ( W ) − ( λ ) for all λ = 0are all smooth and isomorphic, whereas the zero fiber M n (0) is very singular.It is a consequence of deep results on motivic nearby cycles, due to Denef andLoeser (see ), that the virtual motive of the degeneracy locus is related to thedifference of the motives of smooth and singular fibers[ rep n R W ] virt = L − mn ([ M n (0)] − [ M n (1)]) in Mot µ ∞ C [ L − ]It takes a few words to make sense of this equality.The ring of equivariant mtives Mot µ ∞ C is the ring of motives of pairs ( X, µ k )where X is a reduced variety with an action of the cyclic group µ d of d -th rootsof unity, for some d . The scissor- and factoring-relations only hold in situationscompatible with the group action. Further, we have that [( A n , µ d )] = L n whenever K. Behrend, J. Bryan and B. Szendroi, Motivic degree zero Donaldson-Thomas invariants,Inv. Math. 192, 111-160 (2013)
IRTUAL REPRESENTATION MOTIVES 7 the action of µ d on A n is linear. In the above equality we mean with [ M n (1)] really[( M n (1) , µ d )] where µ d acts by multiplying each of the variables X i . On the otherhand, the action of µ ∞ on M n (0) is trivial.The square root of the Lefschetz motive L is the (equivariant) motive 1 − [ µ ],that is, the difference of the motive of one point with trivial action by two pointswhich are interchanged under the µ -action. To understand this we need a generaltrick on separating variables .Consider two superpotentials W ∈ C h X , . . . , X m i d and V ∈ C h Y , . . . , Y k i d anddenote the fibers of the corresponding Chern-Simons functionals M n ( λ ) = T r ( W ) − ( λ ) ⊂ M n ( C ) ⊕ m and N n ( λ ) = T r ( V ) − ( λ ) ⊂ M n ( C ) ⊕ k and the fiber of the sum superpotential S n ( λ ) = T r ( W + V ) − ( λ ) ⊂ M n ( C ) ⊕ m + k then we have the identity of equivariant motives[ S n (0)] − [ S n (1)] = ([ M n (0)] − [ M n (1)])([ N n (0)] − [ N n (1)])Indeed, we clearly have the ’formal’ identity[ S n ( λ )] = X µ ∈ C [ M n ( µ )][ N n ( λ − µ )]which can be made into a proper identity using the fact that [ M n ( λ )] = [ M n (1)]and [ N n ( λ )] = [ N n (1)] for all λ = 0. Then the above means ( [ S n (1)] = [ M n (0)][ N n (1)] + [ M n (1)][ N n (0)] + ( L − M n (1)][ N n (1)][ S n (0)] = [ M n (0)][ N n (0)] + ( L − M n (1)][ N n (1)]from which the identity of equivariant motives follows.If we apply this to W = X and V = Y , then we have for n = 1 that[ M (0)] − [ M (1)] = [ N (0)] − [ N (1)] = 1 − [ µ ]whereas for the sum potential S = X + Y we easily compute[ S (0)] = 2 L − S (1)] = L − L = [ S (0)] − [ S (1)] = ([ M (0)] − [ M (1)])([ N (0)] − [ N (1)]) = (1 − [ µ ]) Virtual motives are a bit harder to define properly (see for full details). If X isa smooth variety then its virtual motive is a scaled version of the ordinary motive[ X ] virt = L − dim ( X )2 [ X ]and in general the virtual motive depends on the singularities of the variety as wellas on its embedding in Y if it is the degeneracy locus of a functional f : Y ✲ C .To appreciate the importance of the µ d -action, consider the superpotential W = XY − Z . Then M (1) ⊂ C is an affine smooth quadric with motive L + L whereas K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. 170,1307-1338 (2009)
LIEVEN LE BRUYN M (0) ⊂ C is a singular quadric with motive L . The degeneracy locus is a singlepoint, the top of the cone in the zero-fiber. However, we get1 = [ • ] virt ? = L − ( L − ( L + L )) = − L − This is caused by the fact that we didn’t use the µ -action on M (1). If we takethis action on V ( xy − z −
1) into account we can on the open piece where x = 0again eliminate y to get a contribution L ( L − x = 0 we have y a freevariable with z = ± i where the two values are interchanged under the µ -action,so this piece contributes L [ µ ]. In total we get[( M (1) , µ )] = L − (1 − [ µ ]) L = L − L and plugging this info in, we get a genuine identity.5. Brauer-Severis
The fibers of the Chern-Simons functional can be viewed as (trace preserving)representation varieties, for we have M n ( λ ) = trep n T n ( λ ) with T n ( λ ) = T m,n ( T r ( W ) − λ )with T m,n the trace algebra of m generic n × n matrices (that is, we adjoin tothe ring generated by the generic matrices all traces of cyclic words in the genericmatrices) and where we consider the trace preserving representations, that is, if X i is mapped to the matrix A i , then its trace T r ( X i ) must be mapped to T r ( A i ).We can apply everything we did before, replacing rep n R by trep n R , to the ring R = T n ( λ ) in order to get( L n − M n ( λ )] = n X k =1 [ S n,k ( λ )] with S n,k ( λ ) = S n,k ( T n ( λ ))with [ S n,n ( λ )] = [ BS n ( λ )][ GL n ] where BS n ( λ ) = BS n ( T n ( λ )). This time, thefibers of the map ψ : S n,k ( λ ) ✲✲ Gr ( k, n )over a k -dimensional subspace V consists as before of points( φ, v ) = ( (cid:20) φ e φ (cid:21) , (cid:20) w (cid:21) )But this time as the only relation is T r ( φ ( W )) = λ , we have with T r ( φ ( W )) = µ that T r ( φ ( W )) must be equal to λ − µ and the extension e can be arbitrary, givingus the formal equality of motives[ S n,k ( λ )] = L mk ( n − k ) [ Gr ( k, n )] X µ ∈ C [ S k,k ( µ )][ M n − k ( λ − µ )]and as before we can convert this to a genuine identity. This then leads to (see ) Lieven Le Bruyn, Brauer-Severi motives and Donaldson-Thomas invariants of quantized three-folds . Journal of Noncommutative Geometry, 12, 671-692 (2018)
IRTUAL REPRESENTATION MOTIVES 9
Theorem 1.
The virtual motive [ rep n R W ] virt can be computed inductively usingthe identity ( L n − M n (0)] − [ M n (1)]) = [ GL n ]([ BS n (0)] − [ BS n (1)])+ n − X k =1 L mk ( n − k ) [ Gr ( k, n )][ GL k ]([ BS k (0)] − [ BS k (1)])([ M n − k (0)] − [ M n − k (1)]) That is, [ rep n R ] virt can be computed from [ BS k (0)] − [ BS k (1)] for all ≤ k ≤ n . At first sight it might seem that computing [ M n ( λ )] is a lot easier than [ BS n ( λ )]as M n ( λ ) is a hyperplane in affine space M n ( λ ) = V ( T r ( W ) − λ ) ⊂ rep n C h X , . . . , X m i = M n ( C ) ⊕ m whereas BS n ( λ ) is a hyperplane in the generic Brauer-Severi variety BS n ( λ ) = V ( T r ( W ) − λ ) ⊂ BS n ( C h X , . . . , X m i )Fortunately, Markus Reineke proved that BS n ( C h X , . . . , X m i ) has a concretecellular decomposition, with cells corresponding to sub-trees τ of the free m -arytree consisting of n nodes BS n ( C h X , . . . , X m i ) = ⊔ τ A d ( τ ) of which the dimensions d ( τ ) can be computed explicitly in terms of a right orderingof monomials in the X i corresponding to the nodes of the extended tree ˜ τ wherewe add leaves to all nodes of τ .For example, BS ( C h X, Y i ) = A ⊔ A where the two cells correspond to thesub-trees τ consisting of two nodes (solid edges) with the extended trees (dashednodes) X Y XX ❉❉❉❉❉❉❉❉❉ Y XY Y X Y ④④④④④④④④④ < X < X < Y X < Y and 1 < X < Y < XY < Y with the boxed terms the nodes of the tree. The dimension of the cell is then thesum over the extended leaves of the number of boxed terms which are smaller, thatis, in our examples 2 + 2 + 2 = 6 resp. 1 + 2 + 2 = 5. This can then be used togive an explicit parametrization of the cells. Here, the two cells consist of triples( X, Y, v ) with ( (cid:20) b d (cid:21) , (cid:20) e fg h (cid:21) , (cid:20) (cid:21) ) resp. ( (cid:20) a b d (cid:21) , (cid:20) f h (cid:21) , (cid:20) (cid:21) ) M. Reineke,
Cohomology of non-commutative Hilbert schemes , Alg. Rep. Thy. 8 (2005)541-561 which makes it easy to compute the motive of the hypersurface V ( T r ( W ) − λ ) ineach cell by elimination of variables (we loose n − n variables in going from M n ( λ )to BS n ( λ )).As an example, consider the superpotential W = X Y + Y X with correspondingJacobian algebra R W = C h X, Y i ( XY + Y X, X )For starters, we have[ M (0)] − [ M (1)] = [ BS (0)] − [ BS (1)] = (2 L − − ( L −
1) = L giving [ rep R W ] virt = 1. To calculate [ rep R W ] virt we need [ BS (0)] − [ BS (1)]which we get from adding the contributions of each of the two cells in the genericBrauer-Severi variety. The fiber T r ( W ) − ( λ ) in the first cell is the hyperplane H ( λ ) = V (2 d h + 2 bh + 2 bdg + 2 df + 2 be − λ ) ⊂ A To compute [ H (0)] − [ H (1)] we first consider the open piece where d = 0 on which wecan eliminate f independent of λ , so this piece does not contribute. If d = 0 we have2 bh + 2 be = λ with f and g free variables. On b = 0 we can eliminate e independentof λ so this does not contribute, and on b = 0 we only get a contribution to [ H (0)]with e, f, g and h free variables, so [ H (0)] − [ H (1)] = L . The fiber T r ( W ) − ( λ ) inthe second cell is the hyperplane H ′ ( λ ) = V (2 d h + 2 bd + 2 ab − λ ) ⊂ A (here, f is a free variable). On b = 0 we can eliminate a independent of λ , so thisdoes not contribute to [ H ′ (0)] − [ H ′ (1)]. If b = 0 we have d h = λ (with a and f free variables) giving [ H ′ (0)] − [ H ′ (1)] = L . That is[ BS (0)] − [ BS (1)] = L + L giving [ rep R W ] virt = L Another example
As a (belated) answer to a question, we will compute the virtual representa-tion motives of a speciafic contraction algebra , which in general are 2-generatedsuperpotential algebras R W assigned to divisorial contractions to curves in 3-folds.Michael Wemyss tells me there are reasons to conjecture that the virtual rep-resentation motives [ rep n R W ] virt of such algebras are fully determined by those insmall dimensions n . For more details see his paper and references contained in it.Consider the superpotential W = X + Y with corresponding Jacobian algebra R W = C h X, Y i ( X , Y )By separation of variables, we have for all n that[ rep n R W ] virt = ( L − n ([ M n (0)] − [ M n (1)])) Will Donovan and Michael Wemyss,
Noncommutative enhancements of contractions (2016), arXiv:1612.01687
IRTUAL REPRESENTATION MOTIVES 11 with M n ( λ ) = { A ∈ M n ( C ) | T r ( A ) = λ } . In this m = 1 case, Reineke’s decom-position result concerns a single tree1 A A . . . A n − A n which gives us that BS n ( C h A i ) = A n with parametrization ( A, v ) = ( . . . a . . . a . . . a ... ... . . . ... ...0 0 . . . a n , )But then, for n ≥ BS n ( λ ) = { A | T r ( A ) = λ } = V ( a n + 3 a n − a n + 3 a n − − λ ) ⊂ A n from which we can eliminate a n − independent of λ , whence[ BS n (0)] − [ BS n (1)] = 0 for all n ≥ rep n R W ] virt for all n ≥ rep R W ] virt and[ rep R W ] virt . More explicitly, we have[ M (0)] − [ M (1)] = [ BS (0)] − [ BS (1)] = 1 − [ µ ]and therefore [ rep R W ] virt = L − (1 − [ µ ]) For n = 2 we have BS ( λ ) = V ( a + 3 a a − λ ) ⊂ A On a = 0 we can eliminate a independent of λ , so this piece does not contribute.On a = 0 we have that a is a free variable, but only when λ = 0, so we get[ BS (0)] − [ BS (1)] = L and [ M (0)] − [ M (1)] = L ( L −
1) + L (1 − [ µ ]) from which we obtain that[ rep R W ] virt = (( L −
1) + L − (1 − [ µ ]) ) For n ≥ M n (0)] − [ M n (1)] = L n − (1 − [ µ ])([ M n − (0)] − [ M n − (1)] + L n − ( L n − − M n − (0)] − [ M n − (1)])Similarly, we have for the superpotential W = X d + Y d that [ BS n (0)] − [ BS n (1)] = 0 for all n ≥ d and therefore that the virtual motives [ rep n R W ] virt can be computed from those with n ≤ d − Acknowledgements.
This is the write up of a talk given in the Glasgow algebraseminar. I thank Theo Raedschelders and Michael Wemyss for an enjoyable stayand stimulating discussions (leading to the final example).