Abstract. The -bosonic–fermionic coinvariant rings in types and admit monomial
bases indexed by two-colored lattice paths, with polynomial exponents bounded
by the path heights. We prove these basis formulas by identifying the polynomial
pieces with explicit complete-intersection colon quotients; rational Newton fields
give rectangular bases, while an exterior Gauss identity supplies the matching
bounds. This description persists in weighted moment quotients under coupled
polynomial perturbations, over any commutative ring in which and the nonempty
subset sums of the weights are units. The same boxes describe local factors under
monic covers and bases for opposite-character wreath products; retaining the two
exterior colors gives an equivariant filtration and the polynomially graded character
in type .
1. Introduction
The ordinary Artin basis describes a coinvariant algebra by a rectangular set of
monomials, with bounds in type . With two exterior families, the corresponding bounds
depend on which exterior variables occur; a mark at one index changes the polynomial
degrees available at every later index. We prove that this dependence is measured exactly
by the heights of a two-colored lattice path, establishing the basis conjectures of
Lentfer [1] for the -bosonic–fermionic coinvariant rings of the symmetric and
hyperoctahedral groups. Their dimensions are and , respectively.
To state the path rule, let count the exterior marks at index and put , with .
Unmarked indices give up-steps, single marks give two colors of flat steps, and double
marks give down-steps, so the bases take the form
These bounds arise from a common calculation with moment ideals, which also explains
why the bases survive changes in the weights and lower polynomial coefficients. The two
coinvariant algebras are specializations of two such families.
The proof must account for each exterior pattern separately, since highest-weight
reduction gives spanning under a perturbation without preventing the dimension from
dropping. The polynomial ideal attached to a pattern is a colon ideal; rational Newton
fields identify it as a complete intersection and give its rectangular basis. Creation
operators then pair triangularly with these boxes, proving independence, while an exterior
Gauss identity bounds every prefix and recovers exactly the same dimensions.
The pairing remains nondegenerate under coupled lower-weight perturbations,
and a universal construction carries the bases to the coefficient rings specified
below.
Let be a commutative ring and fix . Throughout, assume
Put
The two exterior families have bidegrees and .
For the first family, fix , positive integers , and nonnegative integers with . Give
weight . Choose polynomials in all the -variables with highest weighted forms
Thus an exponent zero makes the corresponding polynomial equal to one. Let be
generated, for , by
For , set
and
At arrival height , the four polynomial bounds are
A zero bound contributes an empty box.
Theorem 1.1.The following residue classes form a free -basis of ,
Its rank and total-exterior Hilbert polynomial are
where .
Give weight and weight . The associated graded algebra is the homogeneous algebra
obtained by replacing by . Weighted homogenization gives a finite free degeneration over with this special fiber and basis .
For the strict-height family, choose only with highest forms and let be generated by
Its rank is , and its total-exterior Hilbert polynomial is . With exterior weight zero,
its associated graded algebra is the strict homogeneous power algebra.
The homogeneous specializations , in Theorem 1.2 and , , , in Theorem 1.1 recover
the type- and type-bases displayed above.
The weighted formulation records more than persistence of dimension. For separated
monic covers, the subset-sum hypothesis is exactly the condition for finiteness, and the
same homogeneous algebras describe every local factor; over , the roots of a common
polynomial and the ramification of determine these factors when (Section 7). The
exterior patterns also retain representation-theoretic information. Their path bounds give
bases for opposite-character wreath products, and, in type , retaining each entire two-color
space makes the box decomposition -equivariant and determines its polynomially graded
character (Section 8).
2. Polynomial boxes from rational Newton fields
We begin over a field of characteristic zero or greater than , with every . Put . The
polynomial result needed for the exterior basis is the following. For signs and a monic
polynomial of degree , set
Proposition 2.1.Suppose for every . Then
The colon ideal is generated by a regular sequence whose highest forms have degrees
.
For , we construct the colon generators by rational vector fields and obtain the
rectangular basis by deletion. Deleting a variable changes its multiplicity, so the
induction needs a slightly larger class in which the coordinate multiplicities may
vary.
2.1. The Newton fields. Choose nonnegative integers and signs satisfying
Set
Let consist of vectors with numerator satisfying the following conditions, with
hyperplane indices always ordered
and, when ,
Thus a negative hyperplane permits a simple pole only in the weighted normal direction
.
Define
Let be the first with , or if none exists, and put . Define the column by
The first line is a Newton interpolation column, with a denominator for each earlier
negative sign. Its active interval stops before a larger coordinate multiplicity would violate
divisibility. The second line supplies the residues forced at those earlier negative
hyperplanes. Thus the poles are part of the construction; they allow a negative sign to
lower every subsequent degree by one.
For example, take two variables, , and . The columns are
Their determinant is . Weighted contraction cancels the pole and gives the polynomials
and , of degrees and . The determinant calculation below generalizes this subset-sum
factor to every profile.
Proposition 2.2.The columns (7) form a homogeneous -basis of , of degrees . Their
determinant is
where is a product of signs and ratios of nonempty subset sums of the .
Proof.An active or earlier negative index has , giving the coordinate divisibility. A
positive edge cannot leave a finite active interval, by (6); tangency inside follows
by subtracting evaluations of , and an earlier positive vertex supplies a numerator
factor. At a pole joining an earlier negative index to an active one, the two residues
have ratio . At a pole between two earlier negative indices, their weighted active sums
agree and their partial-fraction residues are opposite. Hence every column belongs
to the module.
At a positive hyperplane, a row change makes one determinant row divisible by its
equation. At a negative hyperplane, a row change puts the possible pole in one row
only. The coordinate orders and homogeneity then give (8), initially with possibly
zero. If , Cramer’s rule proves freeness; every replaced-column numerator has the
same local divisibility, so its coefficient is polynomial.
We prove nonvanishing by deleting the first coordinate. For the all-one profile, we
simultaneously prove the following auxiliary assertion; in the expansion of , for , the
coefficients at height are one at the first such index and zero thereafter; coefficients
at larger heights vanish. The coefficient at index is homogeneous of degree . The
induction has two stages. The auxiliary assertion for variables first proves that the
-variable determinant is nonzero. Cramer’s rule then gives polynomial coefficients in
variables, from which we prove the auxiliary assertion at that size. Primes below
denote the suffix , whose indices retain their original names; in particular, .
For a nonnegative first sign, the first row is , and the suffix rows differ from the
suffix matrix by factors . This gives a nonzero determinant factor up to sign. For a
negative first sign, put , factor from the first row and from each suffix row. The
matrix becomes
Write . Since the first suffix column is , we have , so the Schur complement is
. Local divisibility and homogeneity make this expression constant. To evaluate it,
note that
Terms of height less than two tend to zero. Terms of height greater than two have
, and at height two the suffix assertion leaves only the first index , if it exists. Since
, the remaining term tends to . Thus the constant is
if none exists, and is nonzero.
To finish the simultaneous induction, is . For , the suffix coefficient vectors are,
according as the first sign is ,
where is the first coefficient in the full expansion. Suppose . If , then , so and
only remains. If , then , and only the summand with remains. If , then ; the terms
vanish, leaving only . In each case the suffix assertion gives one at the first index of
the required height and zero thereafter. Coefficients with vanish by their negative
degree. This proves the auxiliary assertion and the all-one case. Constant profiles
follow from .
For a nonconstant profile with first sign negative, let and let be the first later
index with . All later multiplicities are at least . The same factoring gives lower-left
block , where for and zero otherwise, and
If , the lower-left block is zero and the Schur complement is one. Otherwise the
initial suffix block has constant multiplicity . Every sign there is nonpositive; a
positive sign would force , contrary to the choice of . Each of its Newton columns is
supported on that block, and its first column is . By the constant-profile case, these
columns expand with polynomial coefficients. Their degrees satisfy , while
The coefficients in the expansion of do not involve . Consequently its contribution
to the Schur complement tends to zero. The first-column contribution from tends to
. The Schur complement is therefore . A nonnegative first sign again strips the first
row. This completes the induction and proves the stated form of . □
2.2. Contraction and deletion. Weighted contraction cancels the polar parts
If , then is a unit. If every , these generators form a finite-colength regular
sequence. Indeed, the polynomial fields belong to the module for , so contains
Its only zero is the origin. Indeed, the first Laurent coefficients of vanish there. Its
denominator has degree at most and its numerator has smaller degree, so this forces the
rational function to be zero. A nonzero repeated value would then give the nonzero
residue , a contradiction.
Proposition 2.3.If every , the monomials with form a basis of .
Proof.Choose the first negative vertex with , or, if none exists, the last vertex with
. Deletion lowers by one, preserving (6) and changing only . Restriction sets and
removes ; an incident negative edge changes the remaining multiplicity to , a positive
edge to , and an absent edge leaves it unchanged. For no multiplicity changes, while
for the change is , exactly compensating for the removed sign; hence all surviving
heights remain unchanged.
Restriction of fields has kernel ; dividing a field that vanishes on by removes
exactly one coordinate divisibility condition. The coordinate and normal conditions
put its image in the stated restricted module. These two modules have the same
Hilbert series, since
Thus restriction is onto. Contraction commutes with restriction, so and . Apply
the usual multiplication sequence for to . The known complete-intersection Hilbert
series of and make its kernel quotient have exactly the Hilbert series of . The
displayed inclusion must therefore be equality. This also covers a deleted height zero,
when . We obtain
Induct on the number of variables and then on . Restriction supplies exponent
zero at , and deletion supplies exponents . □
For a constant profile , express in the Newton basis. The polynomial coefficient matrix
has determinant a nonzero scalar times . Contraction compares two finite-colength regular
sequences. The standard Koszul comparison gives
Indeed, the comparison of Koszul resolutions has top map multiplication by . Dualizing
gives an injection of the shifted complete-intersection quotients, because finite-length
modules have no lower Ext over the polynomial ring. Its kernel description is exactly .
This proves the colon identity and its rectangular basis.
2.3. A monic polynomial in place of a power. We will use the following filtered fact
repeatedly. If the highest forms of polynomials form a regular sequence for a positive
grading, then
For one element, a nonzerodivisor highest form makes the highest form of every product
the product of the highest forms. Induct on the sequence using the quotient filtration. This
proves both (10) and regularity of the actual sequence.
Proof of Proposition 2.1. For , the assertion is (9) and Proposition 2.3. For general
, multiply the zero-profile Newton matrix by . Its weighted contractions are
Their highest forms are the constant-profile- contractions, so (10) gives the
regular sequence and the box basis. The coefficient matrix expressing is unchanged by
this diagonal multiplication; its determinant remains a scalar times . The same Koszul
comparison proves . The nonhomogeneous comparison is valid because finite-length
modules still have no lower Ext. □
3. An exterior Gauss identity
The polynomial boxes will give independent classes in the exterior quotient. For the upper
bound, we use a factorization that constrains vectors killed by the mixed contractions. Its
degree bound will become the nonnegative-height condition on every prefix of a
path.
Let and let be a matrix over a commutative ring with invertible. On the exterior
algebra of pairs, use left derivatives and put
Anticommutation gives
like-letter families commute. The transposition in the last summand of allows
nonsymmetric matrices.
Write , , and let send to , with the same convention for .
Lemma 3.1.As a formal identity in , with ,
If has exterior degree and for every , then when . Otherwise
In particular, the left side is polynomial of degree at most .
Proof.For one pair and , both sides of (11) act on by
and act as the identity on . This is the usual two-by-two Gauss factorization; its
diagonal factor explains the determinant and the two exterior substitutions. Taking
tensor products proves the formula for diagonal . Conjugating by replaces by
and fixes , so the identity holds for complex matrices with distinct eigenvalues.
Every formal coefficient is a polynomial in the entries of over ; exterior exponentials
truncate at , and formal inversion introduces no denominators. Zariski density
therefore proves the identity over that ring, and hence over every allowed coefficient
ring.
If is maximal with , the relations give
For , exterior degree gives and . Every nonzero integer factor is a unit, so . For
, directly times the top wedge with coefficient , again giving . Thus and . The last
exponential in (11) kills termwise; comparison of equal powers of , followed by ,
gives (12). □
3.1. A rank-one perturbation. The odd contractions enter the same identity through
a rank-one perturbation of a diagonal matrix. Let and let be columns of operators.
Assume that all entries of commute. Suppose
kill , as do all mixed currents for . For , , each current correction is a sum of products of
these odd contractions, and also kills .
Put . The inverses factor as
The right exterior corrections are
so they kill before the diagonal actions. The determinant contributes
Thus (12) supplies two polynomial identities, its scalar and coefficients. This is the
only perturbation identity needed below.
4. The homogeneous basis
We first prove Theorem 1.1 over for , , . All moments then have weight under the weights
in the theorem.
4.1. Polynomial independence and a Frobenius dual. Set
The substitution is free with basis , . The ordinary Newton box proves that is a finite
graded complete intersection with basis .
For a valid path, put
Taking in (9) and making the free substitution gives a basis
Multiplying the proposed small box by places its distinct monomials inside this
enlarged box. Therefore
Let extract the coefficient of the unique top-degree box monomial. It is a Frobenius
functional; the graded complete-intersection socle pairing is perfect. Let be transposed
multiplication by on . Then
The full dual is the common kernel of the transposed odd and mixed moments in .
This finite dual requires no polynomial differentiation. More explicitly, identifies with .
Under this identification, the kernel of is exactly . The polynomial independence
statement (14) can therefore be used directly to construct functionals separating the
proposed classes. This remains valid when some covering degree is divisible by the
characteristic.
4.2. Triangular creation. To create a mark at index , we want support only at indices
, with the coefficient at supplying the difference factors contributed by that mark in (13).
The polynomial enforces this support; its evaluation at vanishes for every . It
also lets each commutator below be completed to a full moment sum. Define
They preserve . For example, anticommuting with a -annihilator gives
Terms with vanish if included; expansion of the product expresses the result as a
polynomial combination of ordinary moments, hence zero on . Mixed commutators give
polynomial combinations of opposite odd annihilators. The other supercommutators
vanish.
Apply products of these operators to , in a fixed exterior order, obtaining . Each
created index is at most the operator index, so its exterior support lies below in product
Gale order. Its coefficient at is
For a proposed linear relation in one exterior bidegree, choose its smallest nonzero
exterior block in a total colex order refining this support order, and let be that block. All
remaining blocks lie above , whereas is supported at or below . Only the chosen block can
contribute to their pairing. By (14), is nonzero in , and perfectness gives with . Pairing
the relation with is then nonzero, a contradiction. Thus the proposed monomials are
independent.
4.3. Prefix equations and the upper bound. The upper bound uses the opposite
end of the exterior order. Fix a bidegree and let be the coefficient of the largest exterior
term of a nonzero harmonic, using descending interleaved bits . Extract the coefficient of
its exact suffix beyond a prefix of length , leaving a vector in the exterior algebra of that
prefix. A contraction outside the prefix could contribute to this coefficient only from a
source with an additional suffix mark. That source would be larger than the chosen term
and is absent. Hence all prefix odd and mixed annihilators kill the extracted
vector. Its exterior degree is , so its primitive weight in Lemma 3.1 is exactly
.
Apply Lemma 3.1 with
The primitive bound gives . In the diagonal identity and its rank-one perturbation, set
and multiply by . The scalar coefficient gives below; the coefficient gives . Since the latter
identity has zero constant term in , reversing its powers gives the sharper degree bound .
Thus
are polynomials of degrees at most and .
At a -only mark, apply the prefix creator . Its commutators with mixed annihilators are
the opposite odd annihilators. Up to sign, the completed top coefficient is ; a contribution
with would require a source already containing , hence a larger exterior term. Completing
the mark changes its exponent in the diagonal product from zero to . Diagonal
polynomiality therefore makes
a polynomial. Monic division of its numerator by therefore has zero remainder, so
For a -only mark, the analogous creator gives . If , the created vector has negative
primitive weight and is zero, giving the same conclusion.
To bound the space of possible largest coefficients, we turn these necessary equations
into a complete intersection of the required length. The rational expressions in (16) follow
the path directly; an up-step introduces a factor , a flat step leaves unchanged, and a
down-step removes one factor. Monic division records precisely the remainders that must
vanish for these updates to stay polynomial.
Put and start with , . At an unmarked index set
At a single mark divide , impose for or for , and set
At a double mark make the two monic divisions
impose , and set . Finally impose all coefficients of . Denote the resulting ideal by
.
There is one generator at a flat step, two at a down-step, and terminal generators,
giving altogether because is the number of up-steps minus the number of down-steps. A
zero bound makes its single-mark relation one; otherwise the positive weights of the
generators form the multiset , as follows. A single mark gives its corresponding bound. A
double mark gives weights and . The first is its own bound. Match the second with the
latest unmatched up-step reaching height ; that up-step has bound . The unmatched
up-steps reach heights , and their bounds are precisely the weights of the terminal
coefficients.
Modulo these relations,
At a single mark the imposed factor divides , so it kills the remainder also in this
rational identity. Every remains monic. Consequently the identity can be evaluated at any
common zero of the endpoint generators. There , while a nonzero repeated value among
the would contribute the nonzero residue . Hence every , and then every , is zero. The
endpoint generators therefore form a regular sequence, with quotient length
Equations (16) and the single-mark relations show that this ideal annihilates every
largest harmonic coefficient. In the filtration by exterior blocks, a graded piece
of injects into the space of its largest coefficients. That space is contained in
the dual of , and so has dimension at most . Summing over the blocks gives .
Together with the independent monomials, this proves the homogeneous basis
theorem.
Applying the same prefix equations to the harmonic (15) gives . The endpoint length
and (14) force equality
If a path is inadmissible or has a zero bound, the primitive or single-mark argument
makes the colon the unit ideal.
5. The strict-height basis and the path counts
We prove the homogeneous strict family first with . The ordinary algebra is . In a prefix of
length , the zeroth odd contractions restrict the two exterior spaces to the kernels of
Both have dimension , and the induced mixed pairing is nondegenerate; its possible
radical is the line , whose coordinate sum is the unit . In dual bases, mixed contraction is
the standard contraction on pairs. Its primitive weight is now . This removal of one pair
explains the strict bound .
In the rank-one argument take , . For a largest harmonic coefficient, it gives
of degrees at most . The endpoint recursion is the previous one with every replaced by
and every single-mark factor replaced by one. At the end, impose the coefficients of
The leading coefficient of is . Thus the leading terms in (17) cancel. Single marks
contribute degree , double marks degrees , and the terminal coefficients degrees . Matching
down-steps with up-steps gives exactly the multiset .
At a common zero, the rational recursion and (17) give
Subset-sum residues force every . Thus the endpoint ideal is a complete intersection of
length , and prefix polynomiality gives the corresponding upper bound.
For independence, take the constant-one profile with . Its enlarged colon bounds are .
Multiplication by embeds the boxes in the enlarged boxes. The creators are
The extra makes their ordinary anticommutators positive moments. Their top
coefficient is a unit times
The triangular pairing argument proves independence. The endpoint ideal is
again exactly , by inclusion and the box length. Free substitution gives the
bounds and multiplies every box size by . This proves the homogeneous form of
Theorem 1.2.
Counting the boxes. For the paired family, divide each local bound by . At arrival height ,
the up, combined flat, and down weights are
Let count boxes in the first positions by endpoint height. Then
It follows that . Total exterior degree is , and this polynomial is palindromic, giving the
rank and exterior polynomial of Theorem 1.1.
For the strict family, first take . The first step is compulsory and ; thereafter
Reversing endpoint height to exterior degree and restoring the covering factors proves
the count in Theorem 1.2.
6. Deformation and arbitrary coefficient rings
The homogeneous proof uses the ordinary algebra in two ways; its perfect pairing turns
polynomial independence into exterior independence, and its grading controls spanning.
Both survive a lower-weight perturbation, even when the new coefficients couple different
coordinates. We establish these two facts before descending the construction to arbitrary
coefficient rings.
6.1. The ordinary algebra and its pairing. Work first over . For the paired family,
put and . The characteristic-polynomial recurrence for makes every later ordinary
moment a combination of these; the coefficients need only be fixed as the index varies.
Their highest forms are the homogeneous ordinary regular sequence. Therefore (10) gives
where is the ordinary power algebra. Its box is an actual basis of .
The top-box-coefficient functional is still perfect. Ordered by degree in one copy and
complementary degree in the other, its pairing matrix is block triangular; the diagonal
blocks are the graded complete-intersection socle pairings. This proves nondegeneracy
without any condition on lower coefficients.
For a valid path with positive bounds, put
Its highest form is (13). If a nonzero small-box polynomial satisfied , its highest part
would contradict (14) in . Thus the same small box is independent modulo in the actual
algebra.
6.2. Transfer to the exterior quotient. On use the creators
Their supercommutators are the same polynomial combinations of ordinary or
opposite odd moments as before; the ordinary anticommutator contains exactly .
All coefficient operators commute even when the polynomials couple different
coordinates. The support remains triangular and the top coefficient is a unit times .
The perfect pairing and small-box independence therefore prove independence
of (3).
The highest form of every moment is its power-case moment. Hence the power quotient
surjects onto the associated graded quotient. Its basis spans the latter and, by induction
on weight, spans the actual quotient. Independence makes this surjection an isomorphism,
proving Theorem 1.1over .
For the strict family, use and
The ordinary associated graded and Frobenius arguments are identical. Replace
in (18) by in both creators. Their odd anticommutators are positive ordinary moments,
so the strict power boxes again give actual independence and filtered spanning. This
proves Theorem 1.2over .
6.3. Universal bases and colons. The passage to coefficient rings uses a single
universal family. Let
Adjoin independent variables for all allowed lower-weight coefficients in (2), giving
a polynomial ring . There are finitely many such coefficients. Both rings are
reduced and Jacobson, and every closed residue field has characteristic greater than
.
The descent rests on the following elementary principle; specified elements of a finite
module over a reduced Jacobson ring form a basis if they do so on every closed-point fiber.
Indeed, the map from the free module on those elements has zero cokernel by Nakayama,
while every coordinate of a kernel vector lies in all maximal ideals and is therefore
zero.
The homogeneous ordinary and full quotients are finite over ; each weighted piece is
finite, and all pieces above the uniform box bound vanish on every closed fiber, hence
vanish. The principle gives the power bases over . Filtered reduction then makes the
coupled quotients finite over , and the field proofs give their prescribed bases on every
closed fiber. The same principle gives free -bases. Every allowed choice over is a base
change from , proving both main theorems, including nonreduced rings. Since reductions
strictly lower weight, the exact associated graded identifications also survive base
change.
The endpoint colons require one further argument; their multiplication maps must
remain injective under base change. In the paired recursion replace by , by , and the
single-mark factors by or . In the strict recursion replace by and retain (17). For either
family,
holds over every allowed , with the corresponding free box basis. Over a field, the
highest endpoint generators are the power-case regular sequence, so their quotient has the
same length. The Gauss identity applied to the triangular top coefficient gives the
endpoint-to-colon inclusion; box independence forces equality.
For the universal assertion, the endpoint quotient has its box basis over by the same
finite-fiber argument. Multiplication by maps it into the free ordinary algebra; all defining
products vanish on each closed fiber, hence their coordinates vanish in . This matrix is
injective on every closed fiber. Its maximal minors generate one, so a linear combination of
the corresponding adjugates supplies a left inverse. It remains injective after every base
change, proving (19). Inadmissible and zero-bound paths give the unit ideal by the
primitive and single-mark equations.
The pairing
on each colon quotient is perfect. Over a field this follows from the Frobenius pairing on
and the definition of . Its universal determinant is nonzero on every closed fiber, hence is
a unit; therefore perfectness survives every base change. The same argument proves the
universal version of the polynomial boxes (5). In that case the coefficient matrix and its
unit determinant factor also give the colon inclusion directly by the adjugate
identity.
6.4. Flat and formal versions. Clear denominators in the weights. Weighted
homogenization of the polynomials in (2) has coefficients in , gives the original data at ,
and the power data at . The universal theorem gives the same free basis over , proving the
asserted flat degeneration.
There is also a local version in the opposite filtration direction. Over , take formal series
in with
The same paired boxes form a basis of the completed moment quotient, and its
associated graded for weighted order is the power algebra. The strict version requires only
the corresponding condition on and gives the strict boxes. The endpoint colons and
pairings remain exact over .
Indeed, the complete filtered regular-sequence lemma is (10) with lowest forms; cancel
a lowest coefficient relation using the Koszul syzygies of the regular sequence, and iterate.
Weights strictly increase, so the corrections converge. The finite graded quotient makes the
ordinary local quotient finite; equivalently, sufficiently high filtration pieces stabilize
and Nakayama makes them zero. Its top-box functional is perfect by the same
triangular pairing argument. The transposed are jointly nilpotent, so all formal
creators are well-defined. A nonzero lowest class of gives small-box independence;
triangular creation gives full independence, and reduction terminates in the finite
ordinary algebra. The endpoint inclusions and equal box dimensions give the colon
identities.
7. Monic covers and local factors
For separated monic covers, the preceding bases admit both a relative and a local
interpretation. Assume that is monic of degree , each is monic of degree , and are monic
with , so that Theorem 1.1 applies. Monic division will describe the relative basis, and the
roots of will separate the local factors.
The relative description begins with the substitution , which makes free over on , . For
, flat base change gives
The bases (5), followed by monic division, give the actual rectangular basis of this
covered colon. The transition from to monomials is unitriangular. Multiplication by any
prescribed monic prefactor embeds the corresponding smaller box by a unitriangular
maximal minor.
A monic extension of degree also has a perfect relative pairing; extract the coefficient
of . Its matrix in has zeros below the complementary-degree antidiagonal and
ones on it, so its determinant is . Composing these pairings with the ordinary
moment pairing gives a perfect covered Frobenius pairing over any allowed base
ring.
7.1. The exact finiteness condition.
Proposition 7.1.For the common-monic-cover family, with invertible but
arbitrary weights,
The same criterion holds for the separated strict-height covers.
Proof.Sufficiency is the basis theorem. If a subset sum is a nonunit, pass to a
maximal-ideal residue field containing it and then to an algebraic closure . Choose a
root of . The ordinary -moments vanish on the nonconstant line
because . Thus the ordinary quotient surjects onto . The monic cover is faithfully
free, so the covered ordinary quotient is infinite-dimensional as well. It is the
exterior-degree-zero part of the full quotient. A finite -module cannot have such a
fiber. For the strict family use and the positive ordinary moments. □
The converse uses the common-polynomial structure; Theorem 1.1 asserts the basis
under (1) for arbitrary coupled products .
7.2. Roots and ramification. The basis theorem fixes the global rank as the lower
coefficients vary. The local description explains how that rank is distributed among the
roots; the same homogeneous moment algebras reappear, with covering degrees replaced
by ramification orders. Work over and assume (1). The ordinary support consists exactly
of points with for every . Indeed, the rational function has its first Laurent coefficients
zero at an ordinary zero, and hence is zero. A repeated value with would have nonzero
residue.
Fix such a point and partition indices into . Let be the root multiplicity of at , and
put
Then .
Theorem 7.2.The local factor at is, as an exterior-bigraded algebra, the super
tensor product over of the homogeneous paired algebras with data
The full algebra is the finite direct product of these local factors.
Proof.Choose formal coordinates with , by extracting a root of its leading coefficient
and formal unit. Write
where is a unit. Change odd coordinates to , .
Every old ordinary moment is a sum of homogeneous cluster moments with
coefficients that are common functions of . Conversely, the old ordinary local algebra
is Artinian. For a sufficiently large , Hermite interpolation supplies a polynomial with
jet at a selected root and zero jets at the others, modulo their th powers. Inserting
it as a linear combination of old moments gives the selected homogeneous cluster
moment. Thus the ordinary local ideals agree.
Modulo this common ideal, functions of the nilpotent are represented by finite
polynomials. After the odd coordinate change, the odd coefficients are and , while
the mixed coefficient is . Interpolation with jets separates the odd moments; jets
separate the mixed ones and remove the unit. The full completed ideals therefore
agree. Each cluster quotient is finite and supported at zero, so completion does
not change it. The final product decomposition is the ordinary Artinian idempotent
decomposition; its idempotents are central even elements. □
If , the local rank is
For fixed , the ramification orders over its preimages under sum to . Summing local
ranks and using recovers the global rising-factorial formula. The exterior polynomial
follows with the same factor .
When all roots and preimages are simple and , , each cluster is a strict-height weighted
algebra tensored with one free odd generator. Indeed, in its homogeneous coordinates put
Then . Positive moments change only by times ordinary moments, and mixed
moments by moments times . The remaining equations are precisely the strict equations
in the , and no equation involves .
8. Coinvariants and the color action
8.1. Opposite-character wreath products. Over , let , , and let act with cyclic
characters on , while permutes indices. Its positive invariant ideal is generated by the
sums of the one-index blocks
To see this, average first over the diagonal cyclic group. Multiply the resulting
one-index block sums and subtract terms with coincident indices, which have fewer
occupied indices. Induction gives every nonzero permutation orbit sum. Repeated odd
blocks have zero orbit sum; all other coefficients are nonzero integers and may be divided
out.
Its dimension is , its total-exterior polynomial is , and its full polynomial/exterior
Hilbert series is
For , , both odd families carry the signed permutation representation, and writing
gives and . These are exactly the type- bounds in [1, Definition 9.5], identifying the
specialization of (20) with the proposed basis.
If both cyclic exterior characters are zero, the invariant ideal is instead the strict family
with and . Its basis has and , with dimension . At this is the type-A basis. The type-A
statement holds over ; the wreath-product statements also hold over fields of characteristic
greater than containing a primitive th root of unity, with invertible. These group-action
conditions are separate from the moment theorem, which permits covering degrees
divisible by the characteristic.
8.2. An equivariant filtration in type A. Although the monomial basis distinguishes
the two colors, its path bounds depend only on the number of marks at each index.
Repeating the prefix argument while retaining each entire color space therefore gives a
filtration invariant under mixing the two exterior families, and the polynomial boxes
determine its character.
Let be the strict quotient with and over . Put , , and let mix the two exterior colors
at every index. For , put
Call admissible if all . The prefix bounds give otherwise; for admissible ,
Give both colors at index weight , and give polynomial variables weight zero. The
multiplicity weight distinguishes multiplicity vectors but not their colors. Since ,
comparing these weights is the same as comparing multiplicities from the largest index
downwards. Thus an extra mark outside a prefix still gives a strictly larger block, as
required by the prefix argument.
Theorem 8.1.For smallest-weight initial forms,
Consequently there is a -equivariant flat algebra degeneration with special fiber
The multiplication is induced from the polynomial and exterior products. The special
fiber has an action of .
Proof.Work in fixed polynomial and exterior degrees. Take a harmonic whose largest
multiplicity vector is , and extract any exact colored suffix beyond a prefix. An outside
contraction could come only from a larger multiplicity source, which is absent. The
prefix proof therefore applies to every colored coefficient in this block and places it
in ; inadmissible blocks vanish.
Let be the coefficient space of smallest-weight initial relations in block . A relation
beginning at weight pairs with a harmonic supported at weights at most only
through that block. Orthogonal-complement linear algebra therefore identifies with
the block image of those harmonics; explicitly, . Hence
The basis theorem makes the sum of these dimension bounds an equality, in each
polynomial degree; there are colorings, each with the same box. Every block bound
is therefore equality, proving (21).
The right side is an ideal; adding a nonvanishing exterior factor at multiplies by
, so . Weight scaling commutes with the diagonal color action, and the weight-Rees
family is finite free over . Each block is also stable under independent color changes
at the indices, giving the larger action on the special fiber. □
Semisimplicity of rational representations gives, in each polynomial degree, a generally
noncanonical module decomposition
where and . This is a module decomposition; the algebra structure is the filtered one
above.
The multiplicity of in is . Hence the multiplicity polynomial of is
Equivalently, the full polynomially graded color character is
The index filtration does not supply a simultaneous -equivariant splitting.
8.3. The Eulerian character. Define the homogeneous Eulerian polynomial
Then the ungraded color character has the compact form
Indeed, after the compulsory first up-step write the height as . The up, colored-flat and
down weights in (22) at are , and . Their transfer operator is , so the character is .
Inserting the largest letter in a permutation gives
Substitution , identifies this recurrence with , proving (23).
References
[1]
J. Lentfer, A conjectural basis for the -bosonic-fermionic coinvariant ring, Algebr. Comb. 8
(2025), no. 3, 711–743. DOI 10.5802/alco.424.