Co-quasi-invariant spaces for finite complex reflection groups
aa r X i v : . [ m a t h . C O ] O c t CO-QUASI-INVARIANT SPACESFOR FINITECOMPLEX REFLECTION GROUPS
J.-C. AVAL AND F. BERGERON
Abstract.
We study, in a global uniform manner, the quotient of the ring of poly-nomials in ℓ sets of n variables, by the ideal generated by diagonal quasi-invariantpolynomials for general permutation groups W = G ( r, n ). We show that, for each suchgroup W , there is an explicit universal symmetric function that gives the N ℓ -gradedHilbert series for these spaces. This function is universal in that its dependance on ℓ only involves the number of variables it is calculated with. We also discuss the com-binatorial implications of the observed fact that it affords an expansion as a positivecoefficient polynomial in the complete homogeneous symmetric functions. Contents
1. Introduction 12. Our context 23. General results 64. Formulas for low degree components 85. Colored quasi-symmetric polynomials 96. Proofs 127. Open problems 17References 17 Introduction
For rank n classical families of finite complex reflection groups W , we contribute to thedescription of the diagonal co-quasi-invariant space Q W for W , in several (say ℓ ) sets of n variables. Here, the use of the term diagonal refers to the fact that W is considered asa diagonal subgroup of W ℓ , acting on the ℓ th -tensor power R ( ℓ ) n of the symmetric algebraof the defining representation of W . Instead of the usual one, the action consideredhere is the so-called Hivert-action. Invariant polynomials under this action are known asquasi-invariants (or quasi-symmetric for the symmetric group). Our space Q ( ℓ ) W is simplythe quotient of R ( ℓ ) n by the ideal generated by constant-term-free quasi-invariants for W . Under the same name, an entirely different notion has been considered in [7, 8]. However, the termi-nology of quasi-symmetric polynomials being well ingrained, it seems awkward to call their generalizationto other reflection groups by any other name then quasi-invariant.
We show that the associated multigraded Hilbert series, denoted Q ( ℓ ) W ( q , . . . , q ℓ ) (whichis symmetric in the q i ), can be described in an uniform manner as a positive coefficientlinear combination of Schur polynomials Q ( ℓ ) W ( q , . . . , q ℓ ) = X µ c µ s µ ( q , . . . , q ℓ ) , (1.1)with the c µ independent of ℓ , and µ running through a finite set of integer partitionsthat depend only on the group W . This a typical phenomena in many similar situationssuch as considered in [6]. It has the striking feature that we can give explicit formulasfor the dimension of Q ( ℓ ) W for all ℓ . To see why this is so striking, it may be worthwhileto recall that, for the entirely analogous context of diagonal co-invariant spaces (i.e. theone corresponding to the usual diagonal action of W on R ( ℓ ) n ), a large body of work hasonly recently settled the special case ℓ = 2, but that we know almost nothing yet for ℓ ≥
3. 2.
Our context
A down to earth description of our context may be given as follows. Consider a ℓ × n matrix of variables X := ( x ij ). For any fixed i (a row of X ), we say that the variables x i , x i , . . . , x in form the i th set of variables . In some instances it is worth simplifyingthis notation, and write X = x x · · · x n y y · · · y n ... ... . . . ... z z · · · z n . Thus, x = x , . . . , x n stands for the first set of variables, y = y , . . . , y n for the secondset, . . . , and z = z , . . . , z n for the last set.With the aim of certain describing polynomials in the variables X , we choose to denoteby X j the j th column of X , for 1 ≤ j ≤ n . We assume the same convention for any ℓ × n matrix of non-negative integers A . Moreover, if the A i are the columns of A , we write A = A A · · · A n . We then consider the monomials X A j j := ℓ Y i =1 x a ij ij , as well as X A := n Y j =1 X A j j . For the monomials X A , who clearly form a basis of space of polynomials R ( ℓ ) n := Q [ X ],the corresponding degree vector : deg( X A ) := X j A j , lies in N ℓ . O-QUASI-INVARIANT SPACES 3
Given r, n ∈ N + , recall that the generalized symmetric group W = G ( r, n ) may bedescribed as the group of n × n matrices having exactly one non zero coefficient in eachrow and each column, which is a r th root of unity. One usually considers W as acting onpolynomials in R ( ℓ ) n = Q [ X ] by replacement of the variables by the matrix X w . Withthis point of view, we may consider that W is generated by the transpositions s j , whichexchange columns j and j + 1 in X , together with s which multiplies the first columnof X by a (chosen) primitive r th root of unity. These generators s j satisfy the usualCoxeter relations for j ≥ s j = Id , ( s j s j +1 ) = Id , and s j s k = s k s j , when | j − k | > . For the special pseudo-reflection s , we have s r = Id and ( s s ) r = Id. This is thediagonal action which is considered for the “usual” definition of the diagonal co-invariantspace for W (see [5]). Rather than this space, we consider a variant below.Our point of departure from the “classical” situation is to consider rather the diagonalquasi-invariant polynomials for W . For example, taking W = G (1 ,
3) (the symmetricgroup S ) and ℓ = 1, we have the quasi-invariant (or quasi-symmetric) polynomials: x + x + x , x + x + x , x + x + x ,x x + x x + x x , x x + x x + x x , x x + x x + x x . For ℓ = 2, another S -quasi-invariant is the polynomial y x y + y x y + y x y . The vector space of diagonally quasi invariant for W = G ( r, n ) is spanned by the mono-mial basis { M A } A ∈B r,n , which are indexed by r -composition-matrices . These are thepositive integer entries ℓ × k matrices, 1 ≤ k ≤ n , having all column sums congruentto 0 mod r , with no column sum actually vanishing. We say that we have an r -matrix if this last condition is dropped. The monomial quasi-invariant associated to such a r -composition-matrix is simply defined as M A := X Y ⊆ X Y A , with Y running over all matrices obtained by selecting (in the order that they appear) k columns of X . We sometimes write M [ A ] for M A ( X ). It is easy to check directly thatthis is indeed a basis. For example, we have M h i = X a
To better analyze the structure of this space, we need to consider the action of the generallinear group GL ℓ on R ( ℓ ) n , defined by( f · τ )( X ) := f ( τ X ) , for τ ∈ GL ℓ . (2.2)Observing that the ideal J = J W is invariant under this action, we conclude that Q ( ℓ ) W inherits a GL ℓ -module structure. Since the ideal J is homogeneous for the vector-degree, Q ( ℓ ) W may be graded by this same vector-degree, i.e.: Q ( ℓ ) W = M d ∈ N ℓ Q W,d , with Q W,d denoting the homogeneous component of degree d of Q ( ℓ ) W . It follows that theassociated Hilbert series , Q ( ℓ ) W ( q ), coincides with the character of Q ( ℓ ) W as a GL ℓ -module.To help the reader parse this statement, let us assume that B is a basis consisting ofhomogeneous elements of Q ( ℓ ) W . This means that, for f ( X ) in B , we have f ( q X ) = q d f ( X ) , (2.3)where q stands for the diagonal matrix q = q . . . q ℓ , and q d := q d · · · q d ℓ ℓ . Thus, an homogeneous f ( X ) is an eigenvector of the linear trans-form q ∗ sending f ( x ) to f ( q X ). Recall here that, by definition, the trace of q ∗ , as afunction of the q i , is the character of Q ( ℓ ) W . Summing up, the Hilbert series of the space,defined by the expression Q ( ℓ ) W ( q ) := X d ∈ N ℓ q d dim( Q W,d ) , (2.4)coincides with the (also usual) definition of the character of the corresponding (polyno-mial) representation of GL ℓ .The point of this last observation is that Q ( ℓ ) W ( q ) is Schur-positive , since Schur functions s µ ( q ) appear as characters of irreducible representations of GL ℓ . Indeed, the decomposi-tion into irreducibles of the polynomial GL ℓ -representation Q ( ℓ ) W ( q ) gives a formula of theform (1.1), with µ running through all partitions for which the homogeneous component Q W,µ is non-vanishing. For example, for the symmetric group, one finds the following
O-QUASI-INVARIANT SPACES 5 expressions for Q n := Q ( ℓ ) S n ( q ) Q = 1 , Q = 1 + s ( q ) , Q = 1 + 2 s ( q ) + 2 s ( q ) , Q = 1 + 3 s ( q ) + 5 s ( q ) + 2 s ( q ) + 5 s ( q ) , Q = 1 + 4 s ( q ) + 9 s ( q ) + 5 s ( q ) + 14 s ( q )+10 s ( q ) + 14 s ( q ) . These examples exhibit the announced striking “independence” with respect to ℓ .Before going on with our discussion, let us introduce another GL ℓ -module which isisomorphic (both as a GL ℓ -module and a W -module) to the space Q ( ℓ ) W . For each ofthe variables x ij ∈ X , consider the partial derivation denoted by ∂ x ij , or ∂ ij for short.For a polynomial f ( X ), we then denote by f ( ∂ X ) the differential operator obtained byreplacing the variables in X by the corresponding derivation in ∂ X . The space S ( ℓ ) W of diagonally super-harmonic polynomials with respect to W -quasi-invariants is simplydefined to be the set of polynomial solutions g ( X ) of the system of partial differentialequations f ( ∂ X )( g ( X )) = 0 , for f ( X ) ∈ J . (2.5)Evidently, we need only consider a generating set of J for these equations to characterizeall solutions. The elementary proof (see [5]) that Q ( ℓ ) W and S ( ℓ ) W are isomorphic relies on thefact that there is a scalar product for which S ( ℓ ) W appears as the orthogonal complementof J W .Following our established conventions, S ( ℓ ) W ( q ) stands for the Hilbert series of thegraded space S ( ℓ ) W . From the above discussion, this is equal to the Hilbert series Q ( ℓ ) W ( q ).The advantage of working with S ( ℓ ) W is that we may present a basis in terms of explicitpolynomials (which give canonical representatives for equivalence classes in Q ( ℓ ) W ).To get a better feeling of how things work out, let us first consider the case W = S and ℓ = 2. We may then check that we have the following bases B d for the varioushomogeneous components H (2)3 ,d of the space H (2)3 = S (2) W . B (2)00 = { } , B (2)10 = {− x + x , − x + x } , B (2)01 = {− y + y , − y + y } , B (2)20 = {− ( x − x ) ( x − x + x ) , − ( x − x ) ( x − x + x ) } , B (2)11 = { x y − x y − x y + x y − y x + y x ,x y − x y + x y − y x − x y + y x } , B (2)02 = {− ( y − y ) ( y − y + y ) , − ( y − y ) ( y − y + y ) } . J.-C. AVAL AND F. BERGERON
Observe that we can calculate B (2)0 k from B (2) k by exchanging all x i by the corresponding y i .By contrast, for ℓ = 3, the space S (3)3 = S (3) W affords the following bases. For all d ofthe form jk
0, we may choose B (3) jk := B (2) jk . To get the bases for the other non-vanishingcomponents of S (3) W , we set B (3) j k equal to the set of polynomials obtained by exchangingthe y i by the corresponding z i for all elements of B (3) jk . In turn, we get B (3)0 jk from B (3) j k ,now exchanging the x -variables for the y -variables. It can then be checked that there areno other non-vanishing component in H (3)3 . One may also use Theorem 3.1 to see this.Our point here is that we get the two Hilbert series H (2)3 ( q , q ) = 1 + 2 ( q + q ) + 2 ( q + q + q q ) H (3)3 ( q , q , q ) = 1 + 2 ( q + q + q ) + 2 ( q + q + q + q q + q q + q q )both taking the form H ( q ) = 1 + 2 s ( q ) + 2 s ( q ), as announced.3. General results
Theorem 3.1.
For any given complex reflection group W = G ( r, n ) , the Hilbert series Q ( ℓ ) W ( q ) affords an expansion in terms of Schur functions, with positive integer coefficientsthat are independent of ℓ , the sum being over the set of partitions of integers d : ≤ d ≤ r n − r − n, (3.1) and having at most n parts. To better underline one of the most important feature of this statement, we mayconsider that the symmetric function involved in these expressions are written in termsof infinitely many variables q = q , q , q , . . . This makes formula (1.1) entirely independent of ℓ . To get the Hilbert series in the specialcase of ℓ sets of n variables, we simply specialize this “universal” formula by setting allvariables q k , for k > ℓ , equal to zero. This process is made even more transparent by“removing the variables”, writing s µ (or h µ ) instead of s µ ( q ) (or h µ ( q )) in (1.1). Inother words, we consider f ( q , . . . , q ℓ ) as the evaluation, of a (variable free) symmetricfunctions f , inn the set of ℓ variables q , . . . , q ℓ . We may also drop the ℓ in Q ( ℓ ) W .The following formula, for the case ℓ = 1, is shown to hold in [1]. Namely, for thegroup W = G ( r, n ), we have Q W ( q, , , . . . ) = (cid:18) − q r − q (cid:19) n · n − X k =0 n − kn + k (cid:18) n + kk (cid:19) q r k . (3.2) As in Macdonald [10], we write h k for the complete homogeneous symmetric functions. O-QUASI-INVARIANT SPACES 7
In particular, for n = 2, we get Q W ( q, , , . . . ) = (1 + q + . . . + q r − ) (1 + q r ) . Since h k ( q, , , . . . ) = q k , this is readily seen to be the specialization at q = q, , , . . . of the following “universal” formula (see [6]), for the groups W = G ( r, r − X k =0 h k ! + r − X k =0 ( k + 1) h r + k + r − X k =1 ( r − k ) h r − k . (3.3)As such, it holds for the diagonal co-invariant space (under the classical action) which,in this very specific case, coincides with the space of diagonal co-quasi-invariant space.A nice feature of the expression given in (3.3) is its h -positivity : X µ a µ h µ , with a µ ≥ . This appears to hold for many other reflection groups, in particular when W is a sym-metric group, leading us to state the following. Conjecture 3.2.
For the symmetric group, the Hilbert series Q n ( q ) is h -positive indegrees smaller than n/ . An immediate consequence of this conjecture is that Q n ( q ) has to have a very specificform, since Q n ( q, , , . . . ) = X µ a µ h µ ( q, , , . . . )= X µ a µ q | µ | = n − X k =0 n − kn + k (cid:18) n + kk (cid:19) q k , (3.4)= X β q χ ( β ) , (3.5)with β running over the set of Dyck paths of height n , and χ ( β ) taking as value the x -coordinate of the first point of the path at height n . The passage from (3.4) to (3.5)is classical. It follows that Proposition 3.3.
Conjecture 3.2 implies that Q n ( q ) affords an expression of the form Q n ( q ) = ≤ n/ X β h µ ( β ) ( q ) , (3.6) with β running over the set of all Dyck paths of height n , and µ ( β ) some partition of theinteger χ ( β ) . Here = ≤ n/ stands for equality in degrees less or equal to n/ . See section 5 for more details.
J.-C. AVAL AND F. BERGERON
As of this writing, we do not have a rule for producing the partition µ ( β ) associatedto β , which would haver to be compatible with the actual values given in (4.4).4. Formulas for low degree components
We discuss now how to get explicit polynomial formulas in the variable n for the coef-ficient of h µ , when µ is a partition of a small enough integer. We restrict the discussionto the case W = S n , but much of it holds in generality. We exploit here the fact that lowdegree homogeneous components of the spaces R n and Q n ⊗ R ∼ S n n are isomorphic, where R ∼ S n n stands for the ring of diagonal quasi-symmetric polynomials. This immediatelyimplies that we have explicit formulas for the relevant homogeneous components of Q n ,since we have the explicit expressions R n ( q ) = (1 + H ( q )) n , and R ∼ S n n ( q ) = 11 − H ( q ) , (4.1)where H ( q ) := P k ≥ h k ( q ). It follows that we may calculate the low degree terms of theHilbert series Q n ( q ), via the expansion(1 + H ( q )) n (1 − H ( q )) = 1 + ( n − h + ( n − h + n ( n − h + ( n − h + n ( n − n − h + n ( n − h h + . . . (4.2)Observe that the coefficients for the various h µ in Q n ( q ) agree with those in the right-hand side of (4.2), whenever n ≥
5. This phenomenon seems to hold for n larger thentwice the order of terms calculated.Let us write k ( µ ) for the number of parts of a partition µ , and denote by d µ := d ! d ! · · · d n !the product of the factorials of multiplicities of parts in µ . Here d i is the multiplicity ofthe part i . We then easily calculate that the coefficient of h µ ( q ), in the right hand sideof (4.2), can be written in the form( n ) k ( µ ) d µ − X ν ( n ) k ( ν ) d ν , (4.3)where the summation is over the set of partitions that can be obtained by removing onepart of µ . As usual, we denote by ( n ) k the product( n ) k := n ( n − · · · ( n − k + 1) . Another way of looking at all this is to say that the coefficient of a given h µ stabilizes toa positive valued polynomial in n , as n grows to be large enough. O-QUASI-INVARIANT SPACES 9
Explicit values.
Explicit calculations give the following h -positive expressions, in thecase of the symmetric group. Q = 1 Q = 1 + h , Q = 1 + 2 h + 2 h , Q = 1 + 3 h + 3 h + 2 h + 5 h , Q = 1 + 4 h + 4 h + 5 h + 4 h + 10 h h + 14 h , Q = 1 + 5 h + 5 h + 9 h + 5 h + 18 h h + 5 h +28 h h + 14 h + 42 h . (4.4)From this it would be tempting (as we did in a first draft of this paper) to conjecturethat Q n is always h -positive, but (as shown by recent calculations of H. Blandin andF. Saliola) this fails at n = 7, indeed we get Q = 1 + 6 h + 14 h + 6 h + 14 h + 28 h h + 6 h (4.5)+42 h h + 14 h + 28 h h + 6 h (4.6)+84 h h + 84 h h − h + 132 h . (4.7)Observe the coefficient of h . For the groups W = G (2 , n ), which is the hyperoctahedralgroup B n , we do have the h -positive expressions Q G (2 , = 1 + 2 h + h + h + 2 h + h , Q G (2 , = 1 + 3 h + 2 h + 3 h + 3 h + h + 3 h h +6 h h + 2 h + 3 h h + 5 h + 6 h + 2 h . Explaining when we do have h -positivity is still somewhat mysterious.5. Colored quasi-symmetric polynomials
In light of the results and conjecture considered above, we think it worthwhile to refor-mulate results obtained in [2] from this new perspective. Indeed, the relevant formulastake a new and much nicer format which gives indirect support to our conjecture, sincethe Hilbert series considered happen to be provably h -positive. This was not noticed atthe time of the writing of [2].Let us consider the subspace of colored quasi-symmetric polynomials of the spaceof diagonal S n -quasi-invariants (in the context of X being ℓ × n matrix of variables).Contrary to our previous presentation, colored quasi-symmetric polynomials are notdefined as invariants. They are rather described in terms of a basis, indexed by “colored These where coined to be the G ( ℓ, n )-quasi-symmetric polynomials (or even B -quasi-symmetric when ℓ = 2) in [2, 4], but this terminology leads to confusion in the present context. composition”. Recall that colored compositions of length p are ℓ × p -matrices C = c c · · · c p c c · · · c p ... ... . . . ... c ℓ c ℓ · · · c ℓp with non negative entries, and such that the associated entries reading word (obtainedby reading column by column from left to right, and each column from top to bottom)avoids the pattern of ℓ consecutive zeros.To each colored composition C , we associate a monomial colored quasi-symmetricfunctions by setting: M C := X Y ≤ i ≤ ℓ Y ≤ k ≤ p x c ik i,a ik (5.1)where the sum is over all choices of a ik such that a ik ≤ a i +1 ,k , when 1 ≤ i < ℓ, and a ℓk < a ,k +1 , for 1 ≤ k < p. For example, we have M h i = X a ≤ b 0) or ,p i + (0 , , for all i . We say that n is the height of β , and that we have a horizontal step s i := ( p i − , p i ) , at level k ≥ 1, if y i = k = y i +1 . The set of Dyck paths is denoted by D n . To any givenDyck path β , we associate the composition ν ( β ) obtained by counting the number oflevel k horizontal steps (ignoring the situation when this number is zero), for k < n . Anexample is given in Figure 1, for the Dyck path β = (0 , , (0 , , (0 , , (1 , , (1 , , (1 , , (2 , , (3 , , (4 , , (4 , , (4 , , (5 , , (6 , , (6 , , (6 , , (7 , , (8 , ✲✻(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) sss sss s s sss s sss s s Figure 1. A Dyck path β with ν ( β ) = 132. Proposition 5.1. The Hilbert series of the quotient C n is given by the formula: C n ( q ) = X β ∈D n h ν ( β ) ( q ) , (5.3) whose dependence on ℓ is entirely encapsulated in the number of variables in q . Observe that if we set all the q i equal to 1, we get a combinatorial expression which isinteresting on its own: X β ∈D n Y k ∈ ν ( β ) (cid:18) k + ℓ − k (cid:19) = 1 ℓ n + 1 (cid:18) ( ℓ + 1) nn (cid:19) , (5.4) in which one may consider ℓ as a variable, hence we actually get a polynomial identity .For integral values of ℓ , equation (5.4) may be proven via a simple bijection on paths.Observe also that both spaces C n and Q n coincide when ℓ = 1.6. Proofs Theorem 3.1. Recall that we are asserting here that there exists a universal ex-pression for the Hilbert series of Q W of the form Q W ( q ) = X µ c µ s µ ( q ) , c µ ∈ N , (6.1)with the sum running over partitions µ of integers d ≤ r n − r − n , each such partitionshaving at most n parts. This restriction on the number of parts follows from the factthat this holds for the whole space R n , of which Q W (or rather S W ) can be consideredas a subspace.In order to prove inequality (3.1), we need to introduce some notations. The sum ofall the entries a r -matrix A , divided by r , is an integer that we denote by w r ( A ). Thisis said to be the r -size of A . To avoid ambiguity, we avoid r -matrices having their lastcolumn vanishing. The number of column of such a r -matrix is its length .Given a r -vector V (a single-column r -matrix), there is a lexicographically largest r -matrix A ( V ) such that • all of the columns of A ( V ) are of r -size 1, • the sum of the columns of A ( V ) is V , • the columns of A ( V ) occur in decreasing lexicographic order from left to right.We denote θ ( V ) the first column of A ( V ), and set ∆( V ) := V − θ ( V ). For V tr = (2 , , A ( V ) = , hence θ ( V ) tr = (2 , 10) and ∆( V ) tr = (0 , , r -vector V thesmallest set, denoted by S ( V ), of r -matrices that contains A ( V ) and that is closed underthe operation that consists in taking sum of consecutive columns. For example, still with V tr = (2 , , S ( V ) := , , , It is a well-known fact that the right-hand-side is actually a polynomial in ℓ . Considering entries from top to bottom. O-QUASI-INVARIANT SPACES 13 For two ℓ -row matrices, A = A · · · A k and B = B · · · B j , the concatenation AB is thematrix having columns AB := A · · · A k B · · · B j . For a general r -composition matrix A = A A . . . A k , we define S ( A ) to be the setobtained by all possible concatenation of matrices successively picked from each of thesets S ( A i ).We now come to the definition of the polynomials G [ A ] := G [ A ]( X ) that are usedto prove (3.1). These are indexed by trans r -matrices , which is to say r -matrices A = A A . . . A k for which there exists 1 ≤ j ≤ k such that w r ( A . . . A j ) ≥ j . It is clear thatany r -composition matrix is trans. Let W = G ( r, n ). Definition 6.1. To a trans r -matrix A , we associate the W -quasi-invariant polynomial, G [ A ] = G A ( X ) recursively defined as follows. • If A is a r -composition, we set G A := X B ∈ S ( A ) M B . • If not, there is a unique column decomposition of A as a concatenation A = B V C, where B a r -matrix (say of length j ), V is a non-zero r -vector, and C is a r -composition. We then set G [ A ] := G [ B V C ] − X θ ( V ) j +1 G [ B ∆( V ) C ] . It should be clear that when A = B V C is trans, then so are both B V C and B ∆( V ) C .Thus, the family G [ A ] is well-defined by induction on the length of A . It is helpful to consider an explicit an example. With r = 2 and n = 4, we computethat G (cid:2) (cid:3) = G (cid:2) (cid:3) − x y G (cid:2) (cid:3) = G (cid:2) (cid:3) − x G (cid:2) (cid:3) − x y (cid:0) G (cid:2) (cid:3) − x G (cid:2) (cid:3)(cid:1) . Since G (cid:2) (cid:3) = M (cid:2) (cid:3) + M (cid:2) (cid:3) + M (cid:2) (cid:3) + M (cid:2) (cid:3) ,G (cid:2) (cid:3) = M (cid:2) (cid:3) + M (cid:2) (cid:3) ,G (cid:2) (cid:3) = M (cid:2) (cid:3) + M (cid:2) (cid:3) ,G (cid:2) (cid:3) = M (cid:2) (cid:3) , we finally get G (cid:2) (cid:3) = x y x y + x x y y + x y x y + x y x y − x y y . Observe that the lexicographic order leading monomial of G (cid:2) (cid:3) is precisely X (cid:0) (cid:1) = x y x y . This is shown to hold in full generality in the following proposition. Proposition 6.2. For any trans r -matrix A , the leading monomial of G A ( X ) is X A . Proposition 6.2 is established through the next two lemmas. Lemma 6.3. For any r -composition matrix A , we have G A = G A ( X − ) , (6.2) writing X − for the alphabet obtained from X by removing its first column of variables Proof. We write A = V C , with V the r -vector corresponding to the first column of A .In view of Definition 6.1, we have G V C = X θ ( V ) G ∆( V ) C + G V C ( X − ) (6.3)which implies (6.2). (cid:3) Lemma 6.4. Let A be a r -matrix of length j , and D a r -composition matrix, then wehave G AD ( X ) = X A G j D ( X ) + (terms < lex X A ) . (6.4) Proof. If A is a r -composition, (6.4) is a direct consequence of Definition 6.1. If not,equation (6.4) is shown to hold by induction on the length of A . Consider the uniquefactorization A = B V C with B is a r -matrix of length j , V a non-vanishing r -vector, and C is a r -composition.We use Definition 6.1 and Lemma 6.3 to calculate that G B V CD = G BV CD − X j θ ( V ) n − j − G B ∆( V ) CD = X B G j V CD − X j − θ ( V ) n − j X B G j ∆( V ) CD + (terms < lex X B )= X B G j +1 V CD + (terms < lex X B )= X B G j V CD ( X − ) + (terms < lex X B )= X B V C G p − D ( X − ) + (terms < lex X B V C )= X B V C G p D + (terms < lex X B V C ) . (cid:3) Proof of Condition (3.1) . The proof is now an easy consequence of the following twoobservations. • Any r -matrix A with w r ( A ) = n is trans. Thus any monomial X A with A a r -matrix and w r ( A ) = n is the leading monomial of in ideal generated by quasi-invariant polynomials for the group G ( n, r ). O-QUASI-INVARIANT SPACES 15 • Any monomial of total degree strictly greater than 2 rn − r − n is the multipleof a monomial X A , with A a r -matrix and for which w r ( A ) = n . Since it is truefor any monomial of such degree, any monomial of degree strictly greater than2 rn − r − n lies in the ideal, whence (3.1). (cid:3) Colored polynomials. The main aim of this subsection is to prove Proposition 5.1.Let us first show that formula (5.4) holds for all positive integer values of ℓ , hence itfollows that we have a polynomial identity. Recall that an ℓ -path is a finite sequence ofpoints p i = ( x i , y i ) in the plane such that p = (0 , 0) and p i +1 = ( p i + ( ℓ, 0) or ,p i + (0 , . We say that we have an ℓ -Dyck path if the path satisfies the further condition that x i ≤ y i for all i . Let us denote by D ( ℓ ) n the set of ℓ -Dyck paths of height n ℓ . It is well-knownthat ℓ -Dyck paths are enumerated by the Fuss-Catalan numbers : D ( ℓ ) n = 1 ℓ n + 1 (cid:18) ( ℓ + 1) nn (cid:19) , (6.5)appearing as the right-hand side of (5.4). To relate this to the left-hand-side of (5.4), weconsider ℓ -colored Dyck paths of height n , which are simply height n Dyck path β whosehorizontal steps, at levels k < n , have been colored by elements of the set { , , . . . , ℓ } .Here, we assume that the colors of steps on a same level are weakly increasing from leftto right, according to the color order 1 < < . . . < ℓ . In other words, the coloring is afunction γ , associating to each horizontal step s i of β , a color γ ( s i ), in such a mannerthat γ ( s i ) ≤ γ ( s i +1 ) , if both s i and s i +1 are horizontal steps, hence inevitably at the same level. Thus ℓ -coloredDyck paths are pairs ( β, γ ), consisting of a path with its coloring.We establish formula (5.4) by building a bijection between ℓ -colored Dyck paths ofheight n , and ℓ -Dyck paths of height n ℓ . Given an ℓ -colored Dyck path ( β, γ ), we denoteby a kj the number of level k horizontal steps of color j , and we iteratively construct apath Φ( β, γ ) = q , q . . . , (6.6)using this data. Starting with q = (0 , a kj ’s as k goes from1 to n − j goes from 1 to ℓ , we successively add to π • a kj horizontal steps of length ℓ , followed by • one vertical step (0 , ℓ -Dyck path of height n ℓ . This transformation is illustrated in Figure 2 with { red , green } as color set (i.e.: ℓ = 2).It is easy to check that Φ is indeed a bijection. (cid:3) ✲✻(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ss sss s ss s Φ ✲✻(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ssss s ssss s ss s ss s s Figure 2. The transformation Φ.With the intention of giving a proof of Proposition 5.1, let us recall the following resultof [2] which generalizes the main result of [3]. Lemma 6.5. A monomial basis of the quotient C ( ℓ ) n is given by the monomials X A suchthat π ( A ) is an ℓ -Dyck path. This last statement uses the following “encoding” of monomials X A in terms of latticepaths. The lattice path π ( A ) is obtained by applying the following construction toentries-reading-word w ( A ) of the ℓ × n exponent matrix A . Starting with the point(0 , a of w ( A ) we add a horizontal steps ( ℓ, , π ( A ) associated to the monomial X (cid:0) (cid:1) = x x y y , whose exponent matrix has entries-reading-word 101201, Observe that each horizontalstep is of length two. One associates variables to each level, namely x i at even level2 ( i − y i at odd level 2 i − 1. We then readout the monomial from the path bythe simple device of associating as exponent of the variable of a level, the number ofhorizontal steps on that level. Proof of Proposition 5.1. For a given Dyck path β , let us consider the set C ( β ) of ℓ -Dyck path π such that there exists a coloring γ with Φ( β, γ ) = π . In formula, C ( β ) = { π | ∃ γ such that Φ( β, γ ) = π } , with Φ as in (6.6). If a k is the number of horizontal steps at level k in β , the choice of ℓ -coloring is equivalent to the choice of a monomial x a k k x a k k · · · x a kℓ kℓ , O-QUASI-INVARIANT SPACES 17 ✲✻ s sss ss s sss ss x y x y x y x Figure 3. Example of π ( A ) for ℓ = 2.with a kj giving the number of steps getting to be colored j , hence a k = a k + . . . + a kℓ . TheHilbert series of the resulting set of monomial is h a k ( q ). Since there is independence inthe choice of colorings at different levels, the Hilbert series of the monomials associatedto ℓ -Dyck paths in C ( β ) is h ν ( β ) ( q ). The fact that Φ is a bijection gives the proof ofProposition 5.1, in view of Lemma 6.5. (cid:3) Open problems The main remaining open question in all the above considerations is to find a explicit(even conjectural) candidate for partitions µ ( β ), one for each Dyck path β , which wouldexplain the h -positive expansion in Proposition 3.3. Naturally, similar questions maybe stated whenever the universal Hilbert series Q W ( q ), for a group W , happens to be h -positive. As discussed in the paper, the resulting entirely combinatorial description ofthe universal Q W ( q ) would give, in one compact formula, the GL ℓ -action characters forall the spaces Q ℓW . References [1] J.-C. Aval , Polynˆomes quasi-invariants et super-coinvariants pour le groupe sym´etrique g´en´eralis´e ,Ann. Sci. Math. 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