Generalized coinvariant algebras for $G(r,1,n)$ in the Stanley-Reisner setting

Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n \times n$ monomial matrices whose entries are $r^{\textrm{th}}$ roots of unity and let $0 \leq k \leq n$ be an integer. Recently, Haglund, Rhoades and Shimozono ($r=1$) and Chan and Rhoades ($r>1$) introduced quotients $R_{n,k}$ (for $r>1$) and $S_{n,k}$ (for $r \geq 1$) of the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ in $n$ variables, which for $k=n$ reduce to the classical coinvariant algebra attached to $G_n$. When $n=k$ and $r=1$, Garsia and Stanton exhibited a quotient of $\mathbb{C}[\mathbf{y}_S]$ isomorphic to the coinvariant algebra, where $\mathbb{C}[\mathbf{y}_S]$ is the polynomial ring in $2^n-1$ variables whose variables are indexed by nonempty subsets $S \subseteq [n]$. In this paper, we will define analogous quotients that are isomorphic to $R_{n,k}$ and $S_{n,k}$.


Introduction
In this paper we will study recently introduced generalizations of the coinvariant algebra attached to the complex reflection group G(r, 1, n) of r-colored permutations of [n] = {1, 2, . . . , n}. In the case r = 1, when this group is the symmetric group S n , the coinvariant algebra is defined as follows. Let x n = (x 1 , . . . , x n ) be a list of n variables and let S n act on C[x n ] by permutation of the variables. The symmetric functions are the corresponding invariant subring It is well known that the ring of symmetric functions has algebraically independent homogeneous generators e 1 (x n ), . . . , e n (x n ) where e d (x n ) is the elementary symmetric function of degree d defined by e d (x n ) = 1≤i 1 <...<i d ≤n Using this we may write I n more succinctly as I n = e 1 (x n ), . . . , e n (x n ) and consequently we have R n = C[x n ] e 1 (x n ), . . . , e n (x n ) .
Note that R n is a graded S n -module. By Chevalley's theorem [3], it is known that as an ungraded module we have that R n is isomorphic to C[S n ], the regular representation of S n . Furthermore, R n has Hilbert series given by Hilb(R n , q) = [n]! q = σ∈Sn q maj(σ) , where maj is the major index statistic on S n . This raises the question of whether there is a vector space basis for R n whose elements x σ are naturally indexed by permutations σ ∈ S n in such a way x σ has degree maj(σ), a question that was affirmatively answered by Garsia [5]. Furthermore, Garsia and Stanton [6] exhibited the coinvariant algebra as a quotient of the polynomial ring C[y S ] where y S = {y {1} , . . . , , y {n} , . . . , y {1,2,...,n} } is a list of variables indexed by the non-empty subsets S ⊆ [n]. In particular, they showed an isomorphism between R n and R n = C[y S ] y S · y T , θ 1 , . . . , θ n , where the generators of the ideal in the denominator come in two types; firstly we have y S · y T for non-empty subsets S, T ⊆ [n] with S ⊆ T and T ⊆ S and secondly we have θ i = S⊆[n],|S|=i y S for 1 ≤ i ≤ n. Furthermore, the quotient R n has a similar Garsia-Stanton basis that not only witnesses the major index statistic on S n , but also the descent statistic.
Recently, Haglund, Rhoades and Shimozono [7] defined a generalization R n,k of R n for any pair of integers 0 ≤ k ≤ n with n ≥ 1. More specifically, they set I n,k = e n−k+1 (x n ), . . . , e n (x n ), x k 1 , . . . , x k n and R n,k = C[x n ] I n,k .
When n = k the quotient R n,n coincides with the classical coinvariant algebra R n . In their paper, they show that as an ungraded S n -module we have R n,k ∼ = C[OP n,k ], where OP n,k is the set of ordered set partitions of [n] with k blocks. Furthermore, there exists a statistic comaj on OP n,k , which reduces to the complement of maj when k = n, and the Hilbert series of R n,k is given by Hilb(R n,k , q) = π∈OP n,k q comaj(π) .
In addition to this, they exhibit a Garsia-Stanton type basis for R n,k exhibiting this fact. In this paper we will prove that these generalized coinvariant algebras also have similar quotients of C[y S ], in the sense that we have an isomorphism between R n,k and R n,k , where R n,k is defined as below. We remark that when n = k the quotient R n,n coincides with the quotient R n introduced by Garsia and Stanton [6]. Definition 1.1. Let 0 ≤ k ≤ n be integers with n ≥ 1. In C[y S ] we define the following ideal: I n,k = y S · y T , θ n−k+1 , . . . , θ n , y S 1 · · · y S k , Finally, set R n,k = C[y S ]/I n,k .
Additionally, we will show that the Garsia-Stanton type basis in Haglund, Rhoades and Shimozono has an equivalent natural basis of R n,k . Our methods are inspired by Braun and Olsen [1], who obtain a similar result when n = k. In theory, our method gives a way to define a descent statistic on OP n,k , although we currently do not have a nice combinatorial interpretation for this. Furthermore, when r > 1 Chan and Rhoades [2] define two generalizations of the coinvariant algebra attached to G(r, 1, n). Similarly to the case r = 1, one of these modules is isomorphic to C[OP n,k is the set of k-dimensional faces attached to the Coxeter complex of G(r, 1, n) which can be thought of r-colored ordered set partitions of some set S ⊆ [n] into k blocks. Again, the grading of these modules is controlled by the comaj-statistic on F n,k and OP n,k , and they describe a Garsia-Stanton type basis in this case as well. Just as for r = 1 we will show an isomorphism between their modules and an appropriate quotient of C[y S ] and show that this Garsia-Stanton type basis has an analogous natural basis of our quotients.
In section 2 we will review the necessary background concerning the classical coinvariant algebra of G(r, 1, n), ordered set partitions and their maj and comaj statistics and a bit of Gröbner theory. In section 3 we will define the appropriate analogues of Definition 1.1 for all r ≥ 1 and deduce the analogues of the Garsia-Stanton bases. In section 4 we will use filtrations on R n,k , S n,k , R n,k and S n,k to deduce the desired isomorphisms. In section 5 we will discuss the multi-graded Frobenius image of the modules in the case of the symmetric group S n . In section 6 we will conclude with some remarks and possible future directions.
2. Background 2.1. Ordered set partitions and G n faces. Fix an integer r ≥ 1 that will be suppressed for the remainder of this paper. Let us introduce G(r, 1, n) which from now on will be denoted by G n . Recall that G n ⊆ GL n (C) is the reflection group consisting of all monomial matrices with entries in {1, ζ, ζ 2 , . . . , ζ r−1 } where ζ = exp(2πi/r) is a primitive r-th root of unity. Matrices in G n can be thought of as r-colored permutations π c 1 1 · · · π cn n where π 1 · · · π n ∈ S n and c 1 , . . . , c n ∈ {0, 1, . . . , r − 1} indicate the colors of the π i . For example, when r = 4, the element  will be interpreted as 3 3 1 1 5 2 2 2 4 0 . We will consider the alphabet A r = {i c : i ∈ [n], c ∈ {0, 1, . . . , r − 1} consisting of letters i with an associated color c. On this alphabet we put an order < that weighs colors more heavily than letters in the following way: Given a word w = w c 1 1 · · · w cr r in A r we define the descent set of w by (2.1) i+1 }, and we will write des(w) := |Des(w)|. The major index of w is given by where c(w) denotes the sum of the colors of w. Given our interpretation of elements of G n as r-colored permutations these definitions directly apply to elements of g ∈ G n . For example, when w = 3 3 1 1 5 2 2 2 4 0 as above we have Des(w) = {2, 3} and maj(w) = (3 + 1 + 2 + 2 + 0) + 4 · (2 + 3) = 28. An ordered set partition of [n] is a partition of [n] with a total order on the blocks. An rcolored ordered set partition is an ordered set partition of [n] with a color associated to each of the elements. For 1 ≤ k ≤ n, let OP n,k be the set of r-colored ordered set partitions of [n] with k blocks. Elements within a block will usually be written in increasing order with respect to <. When r = 4, an example of an element in OP 9,5 is given by Throughout this paper, we will use the ascent-starred model for ordered set partitions. This means that we will write down all the elements in the ordered set partition in the order they appear and use stars to indicate which elements should be grouped together in a block. In this model, the above example will be represented as Alternatively, we can represent this as a pair (g, λ) where g ∈ G n satisfies des(g) < k and λ is a partition with k − des(g) − 1 parts, all at most n − k, that indicates which of the ascents should be starred. In the above example, we would have g = 4 3 2 2 3 2 9 1 6 1 1 0 5 2 7 2 8 1 , which has Des(g) = {4, 6}, so des(g) = 2 and of the 6 ascents, the first, second, fourth and sixth are starred. Representing this as a path from (0, 2) to (2,4), where each starred ascent contributes a step east and each unstarred ascent contributes a step south, we get the following picture so the corresponding partition λ is (3, 2) and our example is represented by (4 3 2 2 3 2 9 1 6 1 1 0 5 2 7 2 8 1 , (3, 2)).
1. In the case n = k this definition actually swaps the notion of maj and comaj on G n . This means that we have to look at ascents rather than descents, but this does not change the combinatorial flavor of the statistic, nor its distribution. 2. In Lemma 2.3 below we will show that when r = 1 this notion of comaj coincides with the comaj-statistic defined in Haglund, Rhoades and Shimozono [7]. However, for r > 1 this notion of comaj is not complementary to the maj-statistic defined in Chan and Rhoades [2], as they use a descent-starred model for ordered set partitions. Furthermore, their comajstatistic includes a sum over the complements of the colors of the ordered set partition, whereas our definition includes simply a sum over the colors of the ordered set partition. However, these definitions do not change the idea behind the ordered set partition and each such choice to be made does not affect the eventual distribution of the statistic. Lemma 2.3. Let r = 1 and let maj((g, λ)) be as in equation (2.4) in [7]. Then we have Proof. Note that maj((g, λ)) equals the sum over all the ascents of σ with respect to the weight sequence (w 1 , . . . , w n ), where w i is the number of completed blocks when reaching the i th element of g. For example, when (g, λ) = (2461357, (1, 1)), corresponding to (24|6|1|357), the weight sequence is given by (0, 1, 2, 3, 3, 3, 4). Now, comaj(g) is the sum over all the ascents of σ with respect to the weight sequence (1, 2, . . . , n). Now, every starred ascent of (g, λ) results in the weights of all the ascents after (and including) that ascent to be decreased by 1. Now remember that λ is a permutation of which each part has at most length n − k, and the stars correspond to the horizontal segments in the left bottom justified Ferrers diagram. Therefore, the number of affected ascents equals (1+the height of the last column) + (2 + the height of the second to last column) + . . . + ((n − k) + the height of the first column) = (1 + 2 + . . . + (n − k)) + |λ|.
We will now introduce a similar model for G n faces. We first recall a definition of Chan and Rhoades [2, Defn. 2.1].
where the numbers in each block are assigned a color in {0, 1, . . . , r − 1}, except for possibly (all) the elements in B 1 .
If the numbers in B 1 are colored, we say that σ has dimension m and if the numbers are uncolored we say that σ has dimension m − 1. We define F n,k to be the set of G n -faces of dimension k.
If the elements of B 1 are uncolored, B 1 is referred to as the zero-block of σ. For example, when r = 3 we have that (14 | 5 2 2 1 3 0 | 7 2 | 6 0 ) ∈ F 7,3 and (1 2 4 2 | 5 2 2 1 3 0 | 7 2 | 6 0 ) ∈ F 7,4 . We can represent elements of F n,k in a way similar to our ascent-starred model for ordered set partitions. Elements σ ∈ F n,k will be represented as triples (Z, g, λ) where Z ⊆ [n] has size |Z| ≤ n − k, g is a word in which each of the letters of [n]\Z appears once and is labeled with a color in {0, 1, . . . , r − 1}, des(g) < k and λ is a partition with k − des(g) − 1 parts that are all at most n − |Z| − k. Here, Z represents of the elements of [n] that belong to the (possibly empty) zero-block of σ, and (g, λ) will yield the remaining block ordered set partition using a similar process to before, by choosing which ascents of g to star according to λ. Note that g might not include every number in [n], but that the same process still works. For example, the two examples above will be represented by ({1, 4}, 5 2 2 1 3 1 7 2 6 0 , (2)) and (∅, 1 2 4 1 5 2 2 1 3 1 7 2 6 0 , (3, 1)) respectively.
The comaj-statistic on F n,k is defined as follows.

2.2.
Generalized coinvariant algebras for G(r, 1, n). Let us now introduce the coinvariant algebras studied in this paper. Recall that GL n (C) acts on C[x n ] := C[x 1 , . . . , x n ] by linear substitutions. Therefore, we can consider subring of invariant polynomials defined by It is a classical result that this ring has algebraically independent homogeneous generators e 1 (x r n ), . . . , e n (x r n ), where e d (x r n ) = e d (x r 1 , . . . , x r n ). Therefore, the coinvariant algebra R n := C[xn] .
Furthermore, the Hilbert series of R n is given by Hilb(R n , q) = g∈Gn q maj(g) .
In [2], Chan and Rhoades define two generalizations of R n for all pairs of integers 0 ≤ k ≤ n with n ≥ 1. Definition 2.6. Let n, k be integers with n positive and 0 ≤ k ≤ n. Let I n,k , J n,k ⊆ C[x n ] be the ideals n , e n−k+1 (x r n ), . . . , e n (x r n ) , (2.3) J n,k = x kr 1 , . . . , x kr n , e n−k+1 (x r n ), . . . , e n (x r n ) , (2.4) and let R n,k and S n,k be the corresponding quotient rings When r = 1 the definitions for J n,k and S n,k specialize to the definition of I n,k and R n,k in Haglund, Rhoades, Shimozono and all results stated for J n,k and S n,k in this paper will also hold when r = 1.
One of their main results [2,Cor. 4.12] is the following: Theorem 2.7. As ungraded G n -modules we have R n,k ∼ = C[F n,k ] and S n,k ∼ = C[OP n,k ], where F n,k is the set of k-dimensional faces in the Coxeter complex attached to G n and OP n,k is the set of r-colored k-block ordered set partitions of [n].

Garsia-Stanton type bases for generalized coinvariant algebras.
Let us recall the Garsia-Stanton type bases for R n,k and S n,k , as introduced by Chan and Rhoades [2, Defns. 5.7 & 5.9]. In order to do so we need the classical Garsia-Stanton basis for R n , indexed by elements g ∈ G n . When g = π c 1 1 · · · π cn n , set d i (g) = #{j ≥ i : j ∈ Des(g)} for the number of descents at or after position i. The descent monomial b g is defined by Now, the following set descends to a C-vector space bases for S n,k [2, Def. 5.7 & Thm. 5.8]: . Furthermore, R n,k has a similar basis E n,k [2, Def. 5.9 & Thm. 5.10] given by all elements of the form Since, |D n,k | = |OP n,k | one might wonder whether there is a natural way to index those basis elements by elements of OP n,k . One way to do so is using our ascent starred model for OP n,k .

2.4.
Gröbner theory. In this section we will recall some notions from Gröbner theory. Let k be a field. A monomial order on k[x n ] is a total order < on the monomials x a 1 1 · · · x an n such that 1. 1 ≤ m for any monomial m; 2. if m and n are monomials with m ≤ n and u is any other monomial, um ≤ un. Given a polynomial 0 = f ∈ k[x n ], the leading monomial LM (f ) is the monomial m such that m has nonzero coefficient in f and such that any other monomial n with this property has n ≤ m. Given We know that as a vector space k[x n ]/I has a basis given by the classes of all monomials m that are not contained in LM (I) [4, p. 248, Prop. 1], which we will call the standard monomial basis for k[x n ]/I. Note that m ∈ LM (I) if and only if m is not divisible by any monomial of the form In our case, we will equip C[y S ] with the graded lexicographical monomial order with respect to the ordering of the variables by y S > y T if |S| > |T | or |S| = |T | and min(S\T ) < min(T \S). For example, for n = 3, this order is given by We remark that only this ordering on the variables is essential, because we will mainly work in homogeneous components of C[y S ]. Therefore, one could use any monomial order with this ordering on the variables instead.
3. The quotients R n,k and S n,k .
We will now define the quotients of C[y S ] that will be isomorphic to R n,k and S n,k . Often we will show a result for the S n,k quotient and then most of the arguments will directly transfer over to the case of R n,k . First, we will fix some notation. Recall that C[y S ] is the polynomial ring in variables indexed by the nonempty subsets S ⊆ [n].
Definition 3.1. Let 0 ≤ k ≤ n be integers with n ≥ 1. In C[y S ] we define the following ideals: where S and T range over all pairs of nonempty subsets S, T ⊆ [n] with S ⊆ T and T ⊆ S, Note that ϕ does not vanish on y S · y T , so ϕ does not induce a ring isomorphism between C[y S ]/ y S · y T and C[x n ]. However, it is well known that C[y S ]/ y S · y T has a basis given by multichain monomials and defining ϕ on this basis yields a G n -module isomorphism between C[y S ]/ y S · y T and C[x n ] 1 .
However, even though we will show that there exist bases for R n,k and R n,k (and S n,k and S n,k ) such that the image of the y-variable basis under ϕ is exactly the x-variable basis, the map ϕ will not define a G n -module isomorphism on these bases, not even in the case of the classical coinvariant algebra. Instead, ϕ is often referred to as the transfer map, indicating the analogy between the Stanley-Reisner quotients and the traditional x-variable quotients.
3.1. An intermediate quotient. Before we can prove our main results we will consider the quotient C[y S ] by the ideal y S · y T , θ n−k+1 , . . . , θ n .
be the Stanley-Reisner ring of the Boolean algebra. It is easy to see that C[B * n ] has a C-basis given by multichain monomials, which are monomials of the form . Before we give the proof, let us illustrate the idea of the proof. Let n = 6, r = 3 and consider y = y 5 {4} y 7 {1,3,4} y {1,2,3,4,6} y 4 {1,2,3,4,5,6} . If we want to write this in the formb (g,d) the underlying permutation of g has to be 4abcd5 with {a, b} = {1, 3} and {c, d} = {2, 6}. Now, note that if ab = 31, then y {4,3} will have exponent at least 1 in b g , either because 3 and 1 have different colors, or because 3 and 1 have the same color, which implies that g has a descent at the second position. Therefore, the underlying permutation is 413265. Now, let c 1 , . . . , c 6 be the colors (of 4, 1, 3, 2, 6 and 5). By the above, we have c 2 = c 3 and c 4 = c 5 . Note that y {1,2,3,4,5,6} has exponent c 6 in b g , hence exponent equivalent to c 6 modulo 3 in b (g,d) . Therefore, since 0 ≤ c 6 ≤ 2, we need c 6 = 1. Equivalently, y {1,2,3,4,6} has exponent equivalent to Similarly, c 2 = c 3 = 0 and c 1 = 2, hence the only option for g is 4 2 1 0 3 0 2 2 6 2 5 1 . Note that in this Proof. Suppose that our multichain monomial is of the form y = y a 1 Note that if y =b (g,d) then the one-line notation of the underlying permutation of g has to be g 1 g 2 . . . g n and all elements that are in the same set (from {g 1 < . . . < g i 1 }, {g i m−1 +1 < . . . < g im } and {g i j +1 < . . . < g n }) need to have the same color. Let these colors be c 1 , . . . , c i j and c i j +1 (the last one appearing only if necessary). Indeed, from the definition, if h ∈ G n and h i and h i+1 have different colors then y {h 1 ,...,h i } has exponent c i − c i+1 or c i − c i+1 + r (depending on whether there is a descent or not) and in both cases this exponent is nonzero, sob (h,d) does not equal y. And if h i and h i+1 have the same color, but h i > h i+1 , then y {h 1 ,...,h i } would appear with exponent r > 0, so again this cannot happen. Therefore, the underlying permutation of g is uniquely determined (if it exists). On the other hand, if such c 1 up to c j (and possibly c j+1 ) exist, they are also uniquely determined, by a backwards inductive argument. Indeed, if S i j = [n], then Y [n] has coefficient 0 modulo r, hence we need c j+1 = 0, and else S i j = [n] and c j has to be the exponent of S i j taken modulo n. Now, suppose c k has been determined, then we will determine c k−1 . It is clear that we need c k−1 − c k ≡ a k−1 mod r, and since c k−1 has to be taken from {0, . . . , r − 1} this gives a unique choice. Now, for this choice of the colors, and the corresponding g, we show that there is a suitable d ∈ Z n ≥0 . Note that by construction,b g = y b 1 Using this we can find a different basis for n−k+1 · · · θ dn nb(g,(d1,...,dn−k,0,...,0)) .
Proof. Order the basisb (g,d) according to the monomial order from above. Note that for each monomial y, the set of monomials y ′ with y ′ ≤ y is finite, since any such monomial y ′ must have deg(y ′ ) ≤ deg(y) and there are finitely many such monomials. Now, if we expandb ′ (g,d) in terms of the basis {b (g,d) } we find that b ′ (g,d) =b (g,d) + lower terms with respect to <. Indeed, suppose g has underlying permutation g 1 · · · g n . Set S i = {g 1 , . . . , g i }. Note that if g i > g i+1 , or c i = c i+1 then necessarily we have that y S i occurs inb g with a positive exponent. Note that (since we only allow multichains), we have θ b a = |S|=a y rb S . Now, terms inb ′ (g,d) correspond to picking one of the terms from each of the θ b a with positive b, in such a way that the result is still a multichain. Because of our monomial order, we should pick from larger a first. Suppose we are picking a subset of size i and suppose i t < i < i t+1 (set i j+1 = n) (we can exclude i = i t , because of the multichain condition we must pick S i ). Then, we are asking for the largest y S with |S| = i and S it ⊆ S ⊆ S i t+1 , which is S = {g 1 , . . . , g i }, due to the fact that g it+1 , . . . , g i t+1 all have the same color and are in increasing order, by the proof of the lemma above. Therefore, the largest possible monomial that could possibly appear is obtained by picking y S i for every i > n − k with d i > 0. Now note that if we take this choice for all i simultaneously we indeed get a multichain monomial, and this monomial is equal tob (g,d) , as desired.
Therefore,b ′ (g,d) expands in a unitriangular way in terms of the basis {b (g,d) } and because of the initial observation in this proof, it follows that {b ′ (g,d) : g ∈ G n , d ∈ Z ≥0 } is a basis for C[B * n ].
Corollary 3.8. Let m be a positive integer. Consider C[y S ]/ y S · y T , θ n−k+1 , . . . , θ n , y S 1 · · · y Sm , where (S, T ) runs over all pairs with S ⊆ T and T ⊆ S, and (S 1 , . . . , S m ) runs over all . This is a finite-dimensional C-vector space with basis given by all elementsb (g,d) with g ∈ G n , d ∈ Z n−k ≥0 and deg(b (g,d) ) < m.

3.2.
Bases for the rings R n,k and S n,k . Note that Corollary 3.8 yields bases for R n,k and S n,k . In this section we will show that these bases can be indexed by elements of F n,k and OP n,k respectively. We will use the models introduced before.
Proof. Let us first show that there is a bijection between elements of the formb (g,λ) andb (g,d) with deg(b (g,d) ) < kr. Note that for any partition λ with parts at most n − k, we have (after extending the above definition to allow for any partition)b (g,λ) =b (g,d) , where d = (d 1 , . . . , d n−k ) with d i = #{j : λ j = i}. Therefore, it suffices to show that λ has at most k − des(g) − 1 parts if and only if deg(b (g,λ) ) < kr. Now, note that if λ has m parts, we have (c i − c i+1 + r · χ(i is a descent)) + c n + mr = c 1 + rdes(g) + mr = c 1 + (m + des(g))r, where χ is the indicator function given by χ(S) if statement S is true and χ(S) = 0 otherwise. Now, since c 1 ∈ {0, 1, . . . , r − 1} we have deg(b (g,λ) ) < kr if and only if m + des(g) ≤ k − 1, that is if and only if λ has at most k − des(g) − 1 parts.
z+1 · · · g cn n and d = (d z+1 , . . . , d n−k ). We now want to show that there is a unique (h, λ) such that (Z, h, λ) ∈ F n,k and b (h,λ) =b (h,d) . However, since deg(b (h,d) ) < kr the first part of the proof shows that indeed we can find such a (h, λ).
Conversely, we show thatb (Z,h,λ) is of the formb (g,d) for a unique (g, d). Write Z = {g 1 < . . . < g z } and let h = h c 1 1 · · · h c n−z n−z . It is clear that we must have g = g c 1 · · · g c z h c 1 1 · · · h c n−z n−z for a suitable c. Furthermore, since we need deg(b g ) ≡ 0 mod r, we in fact have to pick c = 0. Therefore, g is uniquely determined, and hence d (if it exists) is also uniquely determined. By construction, if S t = {g 1 , . . . , g t }, the exponent of y St inb g and inb (Z,h,λ) agree modulo r. Indeed, this is obvious for t > z, and for t ≤ z the choice of c = 0 guarantees this. Furthermore, for t > n − k we still have that the exponents agree as integers (so not only modulo r). Therefore, it only suffices to show that for any 1 ≤ t ≤ n − k the exponent of y St inb (Z,h,λ) is at least the exponent of y St inb g . Again, this is obvious for z < t ≤ n − k. Additionally, it is clear for 1 ≤ t < z, since by construction y St has exponent 0 inb g . Now, for t = z, we are immediately okay if y Sz = y Z occurs with exponent less {0, 1, . . . , r − 1} inb g . Therefore, the only thing that might fail is that y Z occurs with exponent r inb g but exponent 0 inb (Z,h,λ) . However, since deg(b (h,λ) ) < kr, we know that y Z occurs with exponent at least 1 inb (Z,h,λ) and therefore, with exponent at least r, as desired.

A Gröbner theory result.
In this section we will show that the above bases are actually the standard monomial bases with respect to the monomial order used.
Theorem 3.11. Let 0 ≤ k ≤ n be integers with n ≥ 1. Then the set {b (g,λ) : (g, λ) ∈ OP n,k } is precisely the standard monomial basis for S n,k .
Proof. Since we know that the given set is a basis, it suffices to show that the standard monomial basis of S n,k is contained in {b (g,λ) : (g, λ) ∈ OP n,k }.
Similar to Braun and Olsen [1] we show that y ∈ {b (g,λ) : (g, λ) ∈ OP n,k } if and only if y is not divisible by any of the monomials in the list below. The proof will then be completed by showing that each of these monomials occurs as the leading term of some element of J n,k . The list of monomials is given by 1. y S · y T for S ⊆ T and T ⊆ S; 2. y r
Necessity: we will assume that g is of the form π c 1 1 · · · π cn n . Note that if y S with |S| ≥ n − k + 1 occurs in someb (g,λ) then its contribution completely comes fromb g . Now, if there is no descent at position |S|, y S will have exponent c |S|+1 − c |S| ≤ (r − 1) − 0 < r. Furthermore, if there is a descent at position |S|, we have c |S|+1 ≥ c |S| , so y S will have exponent r + c |S|+1 − c |S| ≤ r. Therefore, if y S occurs with exponent at least r, it occurs with exponent exactly r, we have a descent at position |S| and c |S|+1 = c |S| .
Since S 1 ⊆ S 2 ⊆ . . . ⊆ S n , every two variables inb (g,λ) will automatically be indexed by subsets one of which is contained in the other. 2. Since m ≥ n − k + 1, y r [m] would have to come from a descent of g at position m with c m = c m+1 . In order to have a descent we need π m+1 < π m . However, π m ∈ [m], hence π m ≤ m, whereas π m+1 ∈ [n]\[m], so π m+1 ≥ m + 1. 3. This was observed above 4. Suppose such a product y r S · y T actually occurs. Since |S| ≥ n − k + 1, y r S comes from a descent at position |S| with c |S|+1 = c |S| , so π |S| > π |S|+1 . Since {π 1 , . . . , π |S| } = S and {π 1 , . . . , π |T | } = T , we have min(T \S) ≤ π |S|+1 < π |S| ≤ max(S), which is an obvious contradiction. 5. Suppose that such a product occurs. Again, y r T has to come from a descent at position |T | with c |T | = c |T |+1 , hence π |T | > π |T |+1 . Note that π |T | ∈ T \S ⊆ [ℓ], so π |T | ≤ ℓ. Furthermore, π |T |+1 ∈ T , hence in particular π |T |+1 > ℓ, which is a contradiction. 6. Suppose such a triple product occurs. Since |S 2 | ≥ n − k + 1, y r S 2 comes from a descent at position |S 2 | with c |S 2 | = c |S 2 |+1 , so we must have π |S 2 | > π |S 2 |+1 . However, π |S 2 | ∈ S 2 \S 1 and π |S 2 |+1 ∈ S 3 \S 2 , so by assumption we have π |S 2 | ≤ max(S 2 \S 1 ) < min(S 3 \S 2 ) ≤ π |S 2 |+1 . 7. We note that where c n+1 = 0, and χ is the indicator function of the indicated event. Sufficiency: Let m = y S ′ 1 · · · y S ′ t be a monomial not divisible by any of the above mentioned monomials. Then combining properties 1. and 7. we may assume S ′ 1 ⊆ S ′ 2 ⊆ . . . ⊆ S ′ t and t < kr. However, we will rewrite this as m = y t 1 S 1 · · · y tu Su , where S 1 S 2 . . . S u . We will first construct the corresponding g ∈ G n , after which the augmentation λ will follow automatically. Firstly, the underlying permutation of σ will be given by putting the elements of S 1 in ascending order, then the elements of S 2 \S 1 , . . ., the elements of S u \S u−1 in ascending order and finally the elements of [n]\S u in ascending order. Now, we have to assign colors to each of the elements. We will give all elements of S 1 the same color, all elements of S 2 \S 1 the same color, . . ., all elements of S u \S u−1 the same color and finally all elements of [n]\S u the same color. We will assign these colors in reverse order. Firstly, assign color 0 to everything in [n]\S u , then assign color t u to S u \S u−1 , then color t u + t u−1 to S u−1 \S u−2 , . . . and finally color t u + t u−1 + . . . + t 1 to S 1 . Here, everything should be interpreted modulo n. It is an easy check that m =b g · y rv 1 S 1 · · · y rvu Su , where v 1 , v 2 , . . . , v u ≥ 0. Now, let us check that g together with some appropriate λ satisfies the condition that m = b (g,λ) . Firstly, using a similar computation to above, rdes(g) ≤ deg(b g ) ≤ deg(m) < kr, hence des(g) < k, as desired. So, to see that the augmented part corresponds to an appropriate λ we have to check two things, namely that v j = 0 if |S j | ≥ n − k + 1 and that v 1 + . . . + v u ≤ (k − des(g) − 1). For the latter, note that so v 1 +. . .+v u < k −des(g), as desired. For the first part, note that if m = |S j | ≥ n−k +1, then y S j has exponent at most r by condition 3. Therefore, if v j > 0, we need v j = 1, and the exponent of y S j in b g equals 0. In particular, σ m and σ m+1 have the same color and σ m < σ m+1 . Furthermore, note that y S j now has exponent exactly r, so in particular we have S j = [m] by condition 2. Now, we distinguish four cases. j = u = 1 In this case, σ m = max(S 1 ) > m and σ m+1 = min([n]\S 1 ) ≤ m, a contradiction. j = 1, u > 1 In this case, σ m = max(S 1 ) and σ m+1 = min(S 2 \S 1 ). By condition 4. this implies σ m > σ m+1 , a contradiction. 1 < j < u Now, σ m = max(S j \S j−1 ) and σ m+1 = min(S j+1 \S j ), but then σ m < σ m+1 contradicts condition 6.
However, this contradicts condition 5. Therefore, we need to have v j = 0 if |S j | ≥ n − k + 1, completing this part of the proof.
Leading monomials: 1. These monomials are among the generators of J n,k . 2. These monomials are the leading monomials of θ m ∈ J n,k . 3. Write m = |S| and consider y S θ m ∈ J n,k . All monomials in this polynomial are of the form y S · y r T where |T | = m = |S|. Note that all such products have S and T incomparable, except for when T = S. Therefore, modulo J n,k this equals y r+1 S , showing that y r+1 S in fact occurs in J n,k . 4. Write m = |S| and consider θ m · y T . All monomials in this polynomial are of the form y r R · y T where |R| = m = |S|. Modulo J n,k this is equal to R y r R · y T where R runs over all such subsets with R ⊆ T . By assumption, S is the smallest such set with respect to the monomial order, hence y r S · y T is the leading term of this monomial. 5. Let m = |T | and note that J n,k contains y S · θ T which modulo J n,k reduces to R y S · y r R where S ⊆ R and |R| = m. Since T = S ∪ [ℓ] it is clear that T is the lexicographically smallest such set, so this polynomial has leading monomial y S · y r T . 6. Let m = |S 2 | and consider y S 1 y S 3 ·θ m . Similarly, this equals T y S 1 y r T y S 3 modulo J n,k where T runs over all m-element subsets S 1 ⊆ T ⊆ S 3 . By assumption, S 2 is the lexicographically smallest such set, hence y S 1 y S 2 y S 3 can be obtained as a leading monomial. 7. These monomials are among the generators of J n,k . This completes the proof.
Similarly, we have the following result.
Theorem 3.12. Let 0 ≤ k ≤ n be integers with n ≥ 1. Then the set {b (Z,g,λ) : (Z, g, λ) ∈ F n,k } is precisely the standard monomial basis for R n,k .
Proof. Again it suffices to show that the standard monomial basis of R n,k is contained in {b (Z,g,λ) : (Z, g, λ) ∈ F n,k }. We will show that a monomial y belongs to this set if and only if it is not divisible by any of the monomials in the exact same list as before, where we need to change the 7th condition into 7'. y S 1 · · · y S kr+1 where S 1 ⊆ . . . ⊆ S kr+1 . Again we will go through the steps necessity, sufficiency and show that they occur as leading monomials in J n,k .
Necessity: Condition 1 is clearly still satisfied, and conditions 2-6 follow by the exact same argument, since the appropriate monomials y S with |S| ≥ n − k + 1 still have to come from the contribution of g to b (Z,g,λ) , since neither Z, nor λ will affect the exponent of these. For condition 7', we note that b (Z,g,λ) might now have degree kr (when Z = ∅), but will never have degree kr + 1 or more.
Sufficiency: There are two cases to consider. Let y be a monomial not divisible by any of the monomials specified in the list. We will show that y is of the form b (Z,g,λ) . If deg(y) < kr, then we set Z = ∅ and use the same procedure as in Theorem 3.11 to find the appropriate (g, λ). If deg(y) = kr, set Z to be the smallest subset S of [n] (in size) such that y S has positive exponent in y. Let e S be the exponent of S and set y ′ = y/y e S S . Now, set y ′′ to be the same monomial as y ′ where each y T is replaced by y T \S . Since deg(y ′′ ) < kr, the same procedure as in Theorem 3.11 can be used to find appropriate (g, λ) to complete the triple (Z, g, λ).
Leading monomials: For conditions 1-6 the reasoning is exactly the same, since none of them use the multichain generators of I n,k . For condition 7', it again follows immediately since these multichain monomials belong to the generators of J n,k .

4.
A filtration of R n,k and S n,k .
We will now prove the main result of this paper, namely the following.
Theorem 4.1. We have G n -module isomorphisms R n,k ∼ = R n,k and S n,k ∼ = S n,k .

2.
For m ∈ C[x n ] a monomial, we let µ(m) be the partition given by µ(y), where y is the unique multichain monomial with ϕ(y) = m.
and define U ⊲µ and U ⊲µ in a similar fashion. Let V ☎µ be the image of U ☎µ in A, V ☎µ be the image of V ☎µ in A and similarly for the other 2. Now, A and A decompose as G n -modules as respectively. The proof of Theorem 4.1 now follows from the lemma below.
This lemma in turn follows from two other lemmas, for which we need another definition.
Definition 4.4. Let (A, A) = (S n,k , S n,k ) (resp. (A, A) = (R n,k , R n,k )). Given a partition µ d with parts that are at most n we say that µ is 1. admissible if µ has less than kr (resp. kr + 1) parts, n − k + 1 ≤ i ≤ n − 1 occurs at most r times and n occurs at most r − 1 times. 2. semi-admissible if λ has less than kr (resp. kr + 1) parts, has at most r − 1 parts equal to n, but some n − k + 1 ≤ i ≤ n − 1 occurs at least r + 1 times. 3. non-admissible if λ has at least than kr (resp. kr + 1) parts or has at least r parts equal to n.
Note that µ is admissible if and only if there exists a basis elementb with µ(b) = µ. A move is replacing y r S by y r S − θ |S| and cancelling out all non-multichain terms or replacing x r i 1 · · · x r i j by x r i 1 · · · x r i j − e j (x r n ) (for i 1 < . . . < i j ), depending on what setting one is working in. The two main lemmas are now as follows: Lemma 4.5. Let y be a multichain monomial in C[y S ] with µ(y) = µ. Then 1. if µ is semi-admissible or non-admissible, y = 0 in A. 2. if µ is admissible, one can perform a finite number of moves to find the expansion of y in A in terms of the Garsia-Stanton type basis. Additionally, any multichain monomial Y that ever appears in this process has µ(Y ) = µ.
Proof. For 1, note that since µ is semi-admissible or non-admissible, y is divisible by y r [n] , y r+1 S for n − k + 1 ≤ |S| ≤ n − 1 or a multichain monomial of length kr (resp. kr + 1). Since the ideal we quotient out by to get A contains y r+1 S ≡ y S · θ |S| for n − k + 1 ≤ |S| ≤ n − 1, we see that all of y r [n] , y r+1 S and the multichain monomials belong to the ideal, hence y = 0 in A. For the second part, recall the monomial order on C[y S ] from before. Now, consider a monomial y with µ(y) = µ. We claim that if y is not a Garsia-Stanton type monomial we can perform a move and rewrite y as a C-linear combination of smaller monomials y ′ with µ(y ′ ) = µ(y). Indeed, since µ is admissible, a monomial y that is not a basis monomial is this for one of four reasons (by the classification of monomials that are or this form given in Theorems 3.11 and 3.12): 1. y is divisible by y r [t] for some n − k + 1 ≤ t ≤ n − 1. 2. y is divisible by y r S y T for S T , |S| ≥ n − k + 1 and min(T \S) > max(S). 3. y is divisible by y S y r T for S T , |T | ≥ n − k + 1 and T = S ∪ [ℓ] for some ℓ. 4. y is divisible by y S 1 y r S 2 y S 3 for S 1 S 2 ⊆ S 3 , |S 2 | ≥ n−k +1 and max(S 2 \S 1 ) < min(S 3 \S 2 ). In these cases, apply the move, replacing y r [t] , y r S , y r T and y r S 2 respectively. Any monomial that remains after crossing out non-multichain monomials is obtained by replacing this specific variable by some y r R , so it suffices to show that y R is smaller than the replaced monomial. In the first case, |R| = t, so y R < y [t] . In the second case, R is a subset of T of size |S| and since min(T \S) > max(S), S was the subset corresponding to the largest possible monomial over all R, and similarly in the other 2 cases. Therefore, we can rewrite each non-basis monomial in terms of smaller monomials, so at some point we'll be left with only basis-monomials as desired.
For the proof of the second lemma we need the following observation. Note that if y is any monomial in C[y S ] we can still define µ(y), even if y is not a multichain monomial. Now, if y is a nonmultichain monomial, let y ′ be the unique multichain monomial with ϕ(y) = ϕ(y ′ ). We claim that µ(y ′ ) ⊲ µ(y). Indeed, starting from y we can repeatedly replace y A y B (for A and B incomparable) by y A∪B y A∩B . Note that on the µ-level this corresponds to replacing (. . . , |A|, . . . , |B|, . . .) by (. . . , |A ∪ B|, . . . , |A ∩ B|, . . .) which strictly increases the corresponding partition in dominance order (since A and B are incomparable). Note that this local replacement does not change the image under ϕ and since µ increases every time we can only do this finitely many times and so we will end up with some multichain monomial and by uniqueness this is y ′ . Also, we have done at least one replacement, so indeed µ(y ′ ) is strictly larger than µ(y).
The x n -variable analogue of the above lemma is the following. . ., y r Sm respectively, then the moves in this case are replacing i∈S 1 x r i , i∈S 2 x r i , . . ., i∈Sm x r i respectively. Proof. Let y be the multichain monomial associated to m.
For the first case, since λ is non-admissible, y is divisible by either y r [n] or a multichain of length kr (resp. kr + 1). In the first case, e n (x r n ) = x r 1 · · · x r n divides m, hence m = 0 in R n,k . In the second case, let j be an element that is in the smallest S such that y S occurs in the multichain. Then x kr j (resp. x kr+1 j ) divides m and consequently m = 0 in A. In the second case, we have that y is divisible by y r+1 S for some S with n − k + 1 ≤ S ≤ n − 1. Suppose S = {i 1 , . . . , i j }. Then apply the move by replacing x r i 1 · · · x r i j . We can "pull-back" the move to C[y S ], where we replace y r S by y r S − θ |S| , but we do not get rid of non-multichains. Now, any monomial that occurs will contain y S y T for |S| = |T | but S = T , hence would have been removed in the y-setting. However, in the x-setting these monomials remain. However, by the above observation, all of these monomials have strictly smaller µ-partition, as desired.
In the third case, again "pull-back" to the y-setting and do the exact same sequence of moves as in part 2 of the above lemma, but again we do not get rid of non-multichains. Instead, we replace them by multichains with the same image under ϕ and again this will strictly increase the µ-partition.
As an example of this phenomenon, consider r = 2 and S 5,4 . In the y-variable setting, consider y = y 3 {5} y 2 {2,5} y 2 {1,2,3,5} . Note that this is not yet of the formb (g,λ) , for example since the appearance of y {2,5} y 2 {1,2,3,5} violates condition 5 in the proof of Theorem 3.11. Therefore, we apply a step and replace y 2 {1,2,3,5} by y 2 {1,2,3,5} − θ 4 and after getting rid of any monomial that is not a multichain monomial we find that Here, the first monomial isb (5 . Also, note that we started with a monomial with µ-partition (4, 4, 2, 2, 1, 1, 1) and each monomial kept that form. Now, in the x-variable setting we consider Here, the first monomial is a generalized Garsia-Stanton monomial, the second monomial is one we have to perform another step on. Now, the last two monomials have y-monomial y 5 {5} y 2
Proof of Lemma 4.3. Let µ be non-admissible or semi-admissible, then the above lemmas show that V ☎µ /V ⊲µ and V ☎µ /V ⊲µ are both trivial G n -modules. Now, suppose µ is admissible. We need to show that sendingb to b = ϕ(b) (forb a Garsia-Stanton monomial with µ(b) = µ induces a G n -module isomorphism. Let g ∈ G n then we can rewrite g ·b in the Garsia-Stanton basis using the moves from part 2 of Lemma 4.5. Now, since the multichain monomial corresponding to πb is given by πb we can use part 3 of Lemma 4.6 to rewrite πb in the same way in this given basis (viewed as basis for V ☎µ /V ⊲µ ), since all the additional monomials that appear belong to V ⊲µ and hence are 0 in the quotient.

Multi-graded Frobenius series
Recall that the irreducible representations of the symmetric group S n are indexed by partitions λ ⊢ n, and the representation corresponding to λ is typically denoted S λ . We have the Frobenius map Frob from the set of (equivalence classes) of representations of S n to the space of symmetric functions given by linear extension of Frob(S λ ) = s λ (x), where s λ (x) is the Schur polynomial associated to λ in an infinite variable set Note that if µ is a partition with parts at most n and if d is a nonnegative integer, the subspaces U d = span{m : deg(m) = d} and U µ = span{m : µ(m) = µ} are S n -stable subspaces of C[x n ] and C[y S ] respectively, and hence so are their images V d and V µ in S n,k and S n,k (for r = 1) respectively. The graded Frobenius character and multi-graded Frobenius character of S n,k and S n,k are grFrob(S n,k ; q) = ∞ d=0 q d Frob(V d ); grFrob(S n,k ; t 1 , . . . , t n ) = µ t m 1 (µ) 1 · · · t mn(µ) n Frob(V µ ), where the sum is over all partitions with parts at most n, and m i (µ) is the number of parts of µ equal to i. We can determine the graded Frobenius image grFrob(S n,k ; t 1 , . . . , t n ).
Remark 5.2. The Frobenius character map has an analogue for G n as well [2, Section 2.4]. The proofs in Section 4 show that R n,k and S n,k are a refined version of the graded G n -modules R n,k and S n,k , in the sense that grFrob(R n,k ; q, q 2 , . . . , q n ) = grFrob(R n,k ; q); grFrob(S n,k ; q, q 2 , . . . , q n ) = grFrob(S n,k ; q), of which we just explicitly handled the case (S n,k , S n,k ) for r = 1. By finding the graded Frobenius image of C[B * n ] as a G n -module and factoring out n i=1 (1 + x r i + x 2r i + . . .) one can obtain a similar result for general r. Because of the relative importance of the S n case over the case for general G n we decided to only prove the result in the setting of the symmetric group and leave the details for general r to the interested reader.

Conclusion
In this paper we studied quotients R n,k and S n,k of C[y S ] that generalize the coinvariant algebra attached to the complex reflection group G n . Furthermore, we studied bases of these rings that are naturally indexed by the set F n,k of k-dimensional faces in the Coxeter complex attached to G n and by the set of r-colored ordered set partitions with k parts respectively.
When k = n, these basis elements b g of the coinvariant algebra are indexed by r-colored permutations g of [n]. These basis elements have the property that deg(b g ) = rdes(g) + c n (where c n is the color of the last element of the permutation) and deg(b g ) = maj(g), where deg(y) is the degree of a monomial in C[y S ] where we set deg(y S ) = |S|. For general k, and σ an rcolored ordered set partition with k parts, the associated basis element b σ has the property that deg(b σ ) = comaj(σ). Therefore, it algebraically makes sense to define a descent statistic on OP n,k by des(σ) = ⌊deg(b g )/r⌋. This raises the following question. Problem 6.1. Find a combinatorial interpretation for this descent statistic, or possibly a complementary ascent statistic defined via either asc(σ) = (k − 1) − des(σ) or asc(σ) = (n − 1) − des(σ).
Another question that remains unanswered is the analogue of a question asked by Chan and Rhoades [2, Problem 7.1], namely how to generalize this to arbitrary complex reflection groups W ⊆ GL n (C). Just as for r > 1 the combinatorics of our quotient R n,k is controlled by the kdimensional faces of the Coxeter complex attached to G n , one might wonder whether the following exists.
Problem 6.2. For any complex reflection group W ⊆ GL n (C) and any 0 ≤ k ≤ n find a quotient R W,k of C[y S ] whose combinatorics is controlled by the k-dimensional faces of something resembling a Coxeter complex attached to W .
One might hope that answering this question would answer the question posed by Chan and Rhoades, by trying to push R W,k forward using the transfer map ϕ.

Acknowledgements
The author is very grateful to Jia Huang for suggesting the approach in Lemma 3.6, pointing out that this can be used to deduce Theorem 5.1 and for his help clarifying various details. Furthermore, the author would like to thank Brendon Rhoades for suggesting this problem and his feedback on preliminary versions of this paper.