Higher Specht bases for
Springer and -Springer rings
Caius Folkerts
Abstract. We construct higher Specht bases for every Springer and -Springer ring
by a uniform triangular pairing that separates copies of the same Specht module
in the same degree. The pairing also gives simultaneous harmonic decompositions,
making sums and intersections of the defining ideals visible on their tableau
index sets. Removing the power relations produces explicit free bases over the
elementary-symmetric parameters and a finite flat family whose fibers split into
tensor products of Springer and coinvariant rings. The underlying parameter
modules are free integrally, while the prescribed unnormalized higher Specht
families are bases over a field exactly in characteristic zero or characteristic greater
than , independently of the exponent bound.
The graded character of a Springer ring specifies how often each Specht module occurs in
each degree. A higher Specht basis realizes these multiplicities by explicit polynomials
indexed by standard tableaux [5]; proving that the proposed polynomials form a basis
requires distinguishing copies with the same shape and degree. We construct a
triangular pairing that separates these multiplicity vectors for every Springer and
-Springer ring, and that makes their harmonic copies compatible as the defining ideal
varies.
The three-row and hook cases for Springer rings, together with a general nonvanishing
theory, were established in [1]. Our pairing resolves the remaining independence problem
and proves the higher Specht basis conjectures of Chou and Hanada for every
partition. The additional symmetric factors for -Springer rings are introduced in
Section 4.
Fix . Pad with zeros to length , and use the Springer presentation [4]
For standard tableaux of a common shape, let be the cocharge filling, let be the sum
of its labels, and let be its catabolizability type, with the conventions of Section 2.1. Put
Here and are the row and column stabilizers, and the label in each box of
specifies the variable receiving that box’s exponent. The group acts by permuting
variables.
Theorem 1.1.For every partition , the set
is a basis of . For each fixed , its -indexed span is isomorphic to the irreducible
symmetric-group module of shape , with the usual standard-polytabloid action.
The indexing realizes the catabolizability formula for the graded Springer character; the
multiplicity of shape is
This character formula is recorded in [3, (1)]. To turn it into a polynomial basis, we
construct an apolar witness for each indexing tableau and order the pairings
triangularly.
Blasiak’s catabolism insertion produces a tableau of shape from the cocharge labels of
a standard word inserting to [3, Theorem 3.4], and the product of its column
Vandermondes annihilates when . Choosing a column reading of the original tableau gives
the witness exponents two monotonicities; squared norm orders the pairings,
and initial-column sums resolve the ties. The resulting matrix is triangular with
positive diagonal, proving independence even among copies of equal shape and
degree.
The elementary-symmetric multipliers for -Springer rings preserve the extremal
exponent layer on which this order depends. The family, previously established for
two-row partitions at the minimal exponent bound [2], is therefore a basis for
every and every (Theorem 4.1). A compatible choice of witnesses makes the
harmonic Specht copies simultaneous for all Griffin ideals with fixed , so that finite
sums and intersections are governed by unions and intersections of tableau index
sets.
Removing the power bounds leaves the same finite family of higher Specht
polynomials, now a free basis over , answering Griffin’s question for the untruncated
rings [8]. This freeness has a geometric interpretation; the parameters define
a finite flat family with a Springer ring at the origin, and their characteristic
polynomial separates every other fiber into Springer and coinvariant factors.
The parameter freeness holds integrally. For the specified unnormalized higher
Specht polynomials themselves, the coefficient pairing gives characteristic zero or
characteristic greater than as the exact range of fields, independently of the exponent
bound.
2. Column witnesses
2.1. Cocharge conventions.
All tableaux use French diagrams. Rows increase from left to right; columns increase
from bottom to top. The row word reads rows from top to bottom, each from left to right.
Cocharge labels start at zero on entry ; the label increases by one exactly when the next
entry is to the left of the preceding entry in that word. The catabolizability type is
computed by the cyclic insertion algorithm recalled in Section 2.3, applied to the cocharge
labels of the row word.
The tableau is weakly increasing along rows and strictly increasing up columns. The
row assertion follows because the label is nondecreasing as the entry increases. For two
entries in one column, the larger entry lies earlier in the row word. Between the two
entry values there must therefore be an inverse descent, giving a strict increase in
cocharge.
Moreover determines . For each cocharge label, its entries in form a consecutive
interval, and occur in increasing order in the row word. Fill the zero-labelled positions in
row-word order by , then the one-labelled positions, and so on. This recovers
uniquely.
2.2. Annihilation.
A product of column Vandermondes records the column lengths of a tableau.
Elementary-symmetric differential operators detect how many of those columns lie wholly
in a given set of variables. This gives the annihilation statement below, including the
outside powers needed for Griffin rings. It is also the membership assertion in the
harmonic-generator theorem of Rhoades–Yu–Zhao [7, Definition 9, Lemma 11 and
Theorem 15].
Lemma 2.1 (Injective-tableau witnesses).Let , let be a positive integer, and let be an injective filling of the French diagram of with labels in . Write for its set of
labels. Let be the product of the Vandermonde polynomials in its column label sets,
with either choice of overall sign, and put
Then
Here is padded with zeroes to length . These are the elementary and power
relations used below. In particular, when , annihilates the Tanisaki ideal .
Proof.For a column label set of , let be its Vandermonde. Every symmetric differential
operator of positive degree in the variables indexed by annihilates . Indeed its value would
be alternating of degree smaller than , whereas every alternating polynomial is divisible by
. In particular
Fix and set . At most columns of meet . Their total size is at most the sum
of the largest column sizes, . Consequently the columns wholly contained in
contain at least labels. Applying (2.1) separately to each such column shows that
has degree in at most . This proves every claimed elementary annihilation identity.
Finally, each variable of has degree at most , and every outside variable has degree ,
proving the power identities.
All these are identities between polynomials with integer coefficients. Once proved over
, they remain valid after base change to any coefficient field. □
We will also use the elementary inclusion when . Indeed, conjugation reverses
dominance, so the final-column sums of are at least those of . Every elementary
generator in the definition of is therefore also a generator of . Together with
the preceding lemma, this is all the inverse-system information needed for the
proof.
2.3. Column reading and catabolism.
Number rows from zero at the bottom. For standard tableaux of the same shape,
index the filling by the labels of ; is the cocharge label in the box containing
.
The following choice of reading word is useful. Read the columns of from left to right,
reading each column from top to bottom. Call the resulting standard word . It has
insertion tableau , just as the usual row-reading word does. Moreover its cocharge label on
each entry of is the same . Indeed, for consecutive entries of a standard tableau, if is
above , its column is weakly to the left of the column of ; otherwise two strict steps along a
row and a column would put an integer strictly between them. In this case precedes in
both reading words. If is not above , its column is strictly to the right, and it follows in
both words. The inverse descents, and hence all cocharge labels, agree. Thus
the new cocharge word is obtained by reading the entries of in this column
order.
Apply Blasiak’s catabolism insertion algorithm to this cocharge word. Keep the original
word positions fixed, scan cyclically from right to left, and cross out a position when it is
accepted. A current label is accepted in row if adding a box there preserves the partition
shape; otherwise increment its label and continue scanning. When a box is added, fill it
with the label of the corresponding box of . Denote the resulting filling by . Blasiak’s
theorem gives
Here we use the algorithm for a standard word and the equality of its output with the
catabolizability type of its insertion tableau [3, Theorem 3.4]; no new insertion-shape
assertion is required.
Let be the row of in , and put
In each column of , order its labels from bottom to top. With the usual choice of signs
for the product of column Vandermondes,
The exponent vectors are distinct, because the entries of in a column of height are
.
Lemma 2.2 (Two monotonicities of the column-reading witness).The vector is
nonnegative and weakly increasing from bottom to top in every column of . Within
a block of equal in such a column, the original column indices in weakly decrease
from bottom to top, while the entries of strictly increase.
Proof.Number the complete right-to-left scans . A letter accepted during scan has
been rejected exactly times, so it is inserted in row . Hence . Every box of is inserted
after the box immediately below it, because all intermediate diagrams are partitions.
Scan numbers therefore weakly increase up each column.
During a single scan the original columns of are processed from right to left.
The boxes in an equal- block of a -column are inserted during that same scan, in
their order from bottom to top. Their original column indices consequently weakly
decrease. Finally, on that block , and strictly increases. □
3. Triangular apolar pairings
The order needed for triangularity comes from the coefficient equation itself. If
the desired exponent is and a witness has distinguished exponent , put . An
exponent contributes to the pairing only when for a witness-column permutation .
Expanding turns this equation into a rearrangement inequality, with its minimum at
.
When the norms tie, only permutations within the equal- blocks remain. The earliest
original columns contain the largest -values in each block, so such a permutation can only
decrease an initial-column sum, whereas the row-prefix minima on the polynomial
side give the opposite inequality. The two bounds resolve the remaining freedom
and lead to the two-stage order below, which will also handle the symmetric
multiples.
Lemma 3.1 (A triangularity criterion).Fix a tableau and let be its first columns.
For exponent vectors put
Let range over distinct nonnegative fillings of one total degree, strictly increasing
in the columns of . Suppose homogeneous polynomials are column-alternating, with
. Assume that every exponent in satisfies . At equality require for every , with
simultaneous equality only when . For each , suppose a witness has distinguished
exponent and column group , fixing any outside labels, such that . Assume is strictly
increasing and weakly increasing in every witness column, and original -column
indices weakly decrease within every equal-block. Define
with a coefficient of negative exponent interpreted as zero. Ordered by increasing
lexicographic , the matrix is upper triangular, with diagonal
For , the ordinary apolar functional is the positive scalar multiple
Proof.A contribution to has . Rearrangement in each witness column gives
Together with the first hypothesis, this gives . If the endpoint norms agree,
strictness of makes preserve the equal- blocks, so . In each such block, consists of
the largest -values, by the column-index hypothesis. Hence
If the endpoint prefix vectors also agree, equality holds separately in each block.
Its -values are distinct, so the largest subset of a given size is unique. Thus preserves
every , and . The second hypothesis also gives . Since and are column-strict, equality
of their column multisets implies . This proves triangularity, including arbitrary ties
in the ordering.
On the diagonal, the contributing permutations are exactly . Column alternation
gives for each , so the two signs cancel. Summing gives (3.1). Ordinary differentiation
contributes the same factorial factor to every summand, proving (3.2). □
Fix a shape , a degree , and one standard tableau of shape . Let be any set of
standard tableaux of shape and cocharge . For each construct as in Section 2.3, and
define the linear functional
Order by the norm and column-prefix key of Lemma 3.1, with , breaking ties
arbitrarily.
Theorem 3.2 (The higher Specht pairing).The matrix
is upper triangular in this order. For each , abbreviate its data by and put . Its
diagonal entry is
Proof.Apply Lemma 3.1 with and . Every monomial exponent is a permutation of ,
so its norm is . If with and , then preserves , and the first entries in each row of are
its smallest. Thus . Equality for every prefix forces equality in each row separately;
each row contributes a nonnegative excess over its smallest possible prefix sum, so
none can cancel another. Successive prefix differences then give , proving .
The same argument applied to forces and then , since is column-strict. Hence
. Cocharge fillings determine their standard tableaux, so the labels are distinct.
Lemma 2.2 supplies the witness hypotheses. Triangularity and the diagonal follow
from (3.1)–(3.2). □
3.1. From one tableau to the Specht copies.
Lemma 3.3 (Multiplicity spaces).Let be a semisimple -module and let be a
primitive idempotent in the shape-block of its group algebra. Independent vectors generate a direct sum of irreducible copies of shape .
Proof.Write the isotypic component as . The image of is a fixed one-dimensional
subspace of tensored with . Independence there is precisely independence of the
corresponding multiplicity vectors, and their group orbits span the asserted sum. □
Take
For each in this set, annihilates ; the witness has shape and . The triangular matrix
therefore descends to , and shows that
is linearly independent for the single fixed standard tableau .
To pass to the complete Specht copies, let be the permutation module on row tabloids.
For a fixed semistandard exponent tableau , the assignment
is well defined on row tabloids and equivariant under . The image of the standard
polytabloid is precisely . Restrict this map to the irreducible Specht module and then
pass to . It is nonzero by Theorem 3.2, hence injective. Its standard-polytabloid basis
maps to exactly the required -indexed family.
For the fixed , all vectors lie in the image of the primitive idempotent , where is the
hook product and . Lemma 3.3 therefore promotes their independence to a direct sum of
the complete -indexed copies.
Distinct shapes belong to distinct isotypic components, and distinct degrees belong to
distinct homogeneous components. The complete set is consequently linearly independent.
Its cardinality equals by the established graded character formula, so it is a basis. The
equivariant maps above prove the stated action on each fixed- subspace. This completes
Theorem 1.1.
3.2. A coincident-shape example.
For , take the fixed tableau and the two tableaux
Both have shape and cocharge five. Their catabolizability types are and , respectively,
and both dominate . Their squared norms are seven and nine, so they occur in that order.
The column-reading witnesses are
with rejection vectors and in the variable order . For the unnormalized Young
symmetrizers used here, the pairing matrix is
The proof requires positivity only on the diagonal; off-diagonal entries may have either
sign. In particular it handles two copies that cannot be distinguished by shape or
degree.
4. Higher Specht bases for Griffin rings
The same pairing accommodates the elementary-symmetric factors in the proposed basis
for Griffin rings. Fix with and . The partition defines the ring, while will denote the
shape of a higher Specht copy and will index its symmetric multiplier. For a subset , put
where is padded by zeroes through length . The ring is
Put and, for a partition of length at most , define
Write for the number of descents of .
Theorem 4.1 (Higher Specht bases for Griffin rings).For every and , the family
is a higher Specht basis of over .
To apply the triangularity criterion, we need to identify the extremal exponents after
multiplication by a symmetric factor and then leave enough room for them in the column
witness. The rectangle bound on will supply exactly that room.
Write and define the boosted tableau by
padding by zeroes through length . Thus the first entries are not boosted. The
resulting tableau is semistandard; both its original cocharge and its boost are
nondecreasing in the entry of , and the original cocharge is strictly increasing in
a column. Equal-cocharge boxes form a horizontal strip, and their entries in
increase in their order from leftmost to rightmost column. Consequently, among
boxes with a fixed base cocharge, the boosts are allocated increasingly by original
column.
4.1. An extremal layer of an elementary-symmetric product.
Use the squared norm and column-prefix sums from Lemma 3.1, on the boxes of
.
Lemma 4.2 (Norm and prefix bounds).Every monomial occurring in satisfies .
If equality holds, then has the same content as and for every . If all these prefix
inequalities are equalities, then . Moreover
Proof.An exponent in is a sum of indicator vectors of subsets , with precisely of
the subsets having size . For any permutation of ,
Also
Both bounds are attained by assigning the largest boosts to the largest entries of
, as in . Expanding proves the norm bound.
Equality in the norm bound forces equality in both displayed bounds. The subsets
are therefore nested, and each consists of largest -entries, with freedom only among
equal values. Thus is a permutation of aligned with ; if , then . For every base value
, the occurrences of receive a fixed multiset of boosts, namely the corresponding
consecutive portion of the sorted boosts. The resulting boosted multisets are strictly
separated; every value in is smaller than every value in if . In particular the total
content is that of .
Since is symmetric, a term can be written , where and with . The permutation
preserves every . Sort increasingly in each original row. This can only decrease
every left-prefix sum. Because the ranges are strictly separated, after this sorting
the boxes occupied by base value are exactly the boxes where in that row. Thus
the first columns select a fixed number of entries from each . Their sum is at least
the sum of that many smallest entries of .
The tableau realizes all these minima simultaneously; equal- boxes have distinct
original column indices, and their assigned boosted values increase in that order. This
proves the prefix bounds. If equality holds for every prefix, equality in the row-sorting
inequalities forces to be row-sorted. Indeed, each row’s sorting excess and each ’s
excess over its smallest selected entries is nonnegative; equality of their total leaves
no cancellation between rows or between base values. Equality for each and every
column threshold then fixes its entry at each equal- box, one at a time. Hence ,
proving .
If , column-strictness forces once . The strict separation of the forces , and then
. The monomial has coefficient one in ; its nested subsets of each prescribed size
are uniquely determined. All contributing row permutations have positive sign, and
there are of them. □
4.2. The extended apolar witness.
The witness splits at the first entries for a specific reason; has at most parts, so those
entries receive no boost. We can keep their Springer column witness unchanged. On every
remaining label, the exponent is the largest permitted by the power relations, and the
rectangle bound on leaves enough room beneath it for the boosted exponent. The
difference from these witness exponents will again be a nonnegative differential
order.
Apply column-reading insertion to , retaining its labels in . This gives an injective
filling of shape on a -element subset . Put
On there is no boost, so is the nonnegative rejection-count vector. Outside , the bound
on gives , so there as well.
Lemma 2.1 gives ; the same membership appears in [2, Theorem 7.8]. The elementary
generators of the ideals give
Indeed the sums of the final columns of are at least those of , so every elementary
generator of is also a generator of ; the power generators are the same. We therefore
obtain the well-defined functional on
Let fix the outside labels. Lemma 2.2 gives all witness hypotheses in Lemma 3.1;
outside labels are fixed singleton blocks.
Proof of Theorem 4.1. Fix a shape, total degree, and one standard tableau .
Lemma 4.2 supplies the polynomial hypotheses of Lemma 3.1, and the preceding
construction supplies its witnesses. The boosted filling determines ; cocharge and
boost both increase weakly with the entry of , so a tie forces equality in both.
Standardizing tied entries in their reading order recovers , and subtracting its
cocharge recovers . Thus the fillings are distinct, as required by the criterion.
The apolar matrix is upper triangular with diagonal
Its functionals annihilate the defining ideal, proving fixed- independence.
Symmetry of preserves the Specht maps and primitive-idempotent images, so
Lemma 3.3 gives independence of the complete copies. The Chou–Hanada monomial
basis and its indexing bijection give the matching dimension [2, Theorem B]; hence
the family is a basis. □
4.3. Other symmetric multiplier bases.
Corollary 4.3 (Dominance-triangular symmetric multipliers).For each auxiliary
partition , choose a homogeneous symmetric polynomial
where the sum is over partitions of strictly dominated by . Replacing by in
Theorem 4.1again gives a higher Specht basis of . In particular one may use the
monomial symmetric polynomials or the Schur polynomials .
Proof.Strict dominance gives . If denotes the entries of in decreasing order,
summation by parts also gives . Consequently every monomial of , for , has squared
norm strictly smaller than . The layer of maximal squared norm in therefore comes
entirely from .
This top layer has exactly the alignment and prefix properties in Lemma 4.2; its
distinguished coefficient is . The apolar matrix remains upper triangular, with each
diagonal multiplied by the nonzero scalar . This proves the corollary. The monomial
case is immediate, and the Schur case uses the usual Kostka expansion . □
5. Simultaneous harmonic bases for the Griffin ideals
To compare different Griffin ideals, the witness for an index must be chosen independently
of the quotient. We use the largest unboosted prefix whose witness fits within the
exponent bound; its harmonic copy then belongs to every quotient in which that index is
eligible. With this common choice, ideal sums and intersections become operations on the
index sets.
For homogeneous polynomials of the same degree, use the bilinear apolar pairing . It is
nondegenerate, and variable permutations are orthogonal. For a homogeneous ideal ,
where equality with the degreewise orthogonal complement uses that is an ideal. These
spaces are closed under differentiation, and under the group algebra when is
-stable. If is a Young symmetrizer, write for its adjoint, obtained by reversing
permutations.
Fix positive integers and . The witnesses can be chosen simultaneously for all
partitions with . The following elementary restriction property identifies the appropriate
common witness.
Lemma 5.1 (Restriction of catabolizability type).If , then is obtained from by
retaining its first boxes in row-superstandard order. In other words, retain successive
complete parts of until fewer than the next part remain, and append that remainder
if it is positive.
Proof.Use the greedy recursive description of catabolizability [3, Proposition 2.2
and the following paragraph]. Let be the largest initial interval lying in the bottom
row of . The first part of is , and the remaining parts are the catabolizability type
of , standardized after removing those entries. For , the restriction is a single row
of size . For , its first greedy catabolism also removes exactly entries. Row insertion
commutes with restriction to an initial alphabet; . Since catabolism is defined by
deleting the initial bottom-row interval and inserting the remaining row words in
the prescribed order, it follows that catabolism also commutes with this restriction.
Apply induction to the standardized tableau and the first of its entries. □
Let consist of pairs with , , and . Trailing zeroes are omitted when writing . For such a
pair write and put
The first bound ensures that the first entries receive no boost, and Lemma 5.1 shows
that the second bound ensures .
Choose one standard tableau of each shape. Construct the column-reading witness on
using the labels of , and use the outside exponent as before. Denote the resulting
polynomial by , its distinguished exponent by , and put . If , the convention is and the
column group is trivial. For the apolar pairing, represents the functional . Applying the
adjoint Young symmetrizer puts this representer in the primitive Specht space of shape .
We therefore define
where is the Young symmetrizer used earlier. These choices depend only on
.
For with , set
The empty partition , , is allowed; its ideal is and its index set is .
Theorem 5.2 (Compatible harmonic copies for fixed ).Each is an irreducible
copy of shape in degree . The same collection gives the simultaneous direct sum
decompositions
Proof.Lemma 3.1 applies to the new witnesses; the retained prefix has no boosts,
its output has at most rows, and all outside boosted entries satisfy . It therefore
proves independence of all in each primitive multiplicity space, for the entire set .
Indeed, with the hook product,
The witnesses lie in the image of the primitive adjoint idempotent , so Lemma 3.3
makes their generated copies irreducible and their sum direct.
All lie in the harmonic space of , which has dimension . Their total dimension is
as well. One direct count is to standardize a word , breaking ties from left to right,
to obtain a permutation . Along the entries of , subtract its cocharge labels from
the corresponding letters of . The differences are nonnegative and weakly increasing;
an inverse descent forces a strict increase of the original letters, compensating for
the cocharge increment. Reversing these differences gives a unique partition . This
operation is reversible, and RSK replaces by a pair of standard tableaux of the same
shape, preserving its cocharge and inverse descents on . Thus the count is exactly .
This proves the decomposition for the empty partition.
Now suppose , and write . The length condition gives . Also implies , so the
restriction lemma gives . Hence and as Young diagrams. The elementary degree
thresholds consequently give
The first inclusion uses dominance at equal size, and the second uses diagram
containment, so every tail sum of is at least the corresponding tail sum of .
Lemma 2.1 now shows that . Thus the displayed direct sum is contained in that
harmonic space. Its dimension equals the dimension of the quotient by Theorem 4.1;
hence it is the entire harmonic space. □
Corollary 5.3 (Sums and intersections).Fix , and take any finite nonempty family
of Griffin ideals , allowing different sizes . The quotient by their intersection has a
higher Specht basis indexed by the union of their sets ; the quotient by their sum
has the analogous basis indexed by the intersection of these sets. In either case the
graded Frobenius character is
where is the appropriate union or intersection. The same statements hold for any
finite expression built from sums and intersections, and one may replace by any of
the multiplier families in Corollary 4.3. In particular the lattice of ideals generated by
these Griffin ideals under sums and intersections is distributive; for any three ideals
in that lattice,
Proof.Orthogonal complements exchange ideal intersections with sums of harmonic
spaces and ideal sums with intersections of harmonic spaces. Theorem 5.2 identifies
these operations with unions and intersections of index sets in a single direct sum. The
selected witnesses annihilate the resulting ideal. Their pairing with the corresponding
candidates is a principal submatrix of the same triangular matrix, with nonzero
diagonal, and the harmonic decomposition supplies the matching dimension. The
Specht-equivariance argument completes the claim. These statements iterate under
further sums and intersections. The two ideal identities follow from the distributive
laws for unions and intersections of their harmonic index sets. □
Remark 5.4 (One partition records all eligibility conditions).For partitions
of possibly different sizes, write if for every . For the canonical partition above,
the condition is exactly . Indeed the row-prefix sums of are ; the height restriction
supplies the cutoff at the first parts, and supplies the unboosted-prefix cutoff.
The same order has the equivalent tail inequalities
This follows from the mutually inverse formulas and . In particular the sum of
two fixed- Griffin ideals is again a Griffin ideal. Its partition is determined by
The right side is a nonnegative, nonincreasing, discrete convex integer sequence,
eventually zero. Its successive differences are therefore the conjugate parts of a
partition of size at most and length at most . It is the least upper bound in , and
the defining elementary thresholds give .
5.1. Springer harmonic spaces. For , the power relations follow from the global
elementary relations, so . The common construction therefore specializes to the following
decomposition, independent of . Write .
Choose once and for all one standard tableau of each shape . For , form the
column-reading witness and rejection vector using . Let be its Young symmetrizer, and
let be the adjoint obtained by reversing every permutation. Put
These polynomials have degree and are defined independently of the partition used for
a quotient.
Corollary 5.5 (Springer harmonic decompositions).Each is an irreducible
-module of shape . The same family of submodules gives, for every partition , the
direct sum decomposition
In particular, all these decompositions occur inside one direct sum decomposition
of .
Proof.Take , , and in Theorem 5.2. Every standard tableau has fewer than descents,
and the canonical prefix is all of . Thus its witness is precisely (5.1), and its eligible
indices are . □
Corollary 5.6 (Finite intersections).Let be a nonempty finite collection of
partitions of , and put
Then
is a higher Specht basis of . Its graded Frobenius character is
Proof.This is the same specialization of Corollary 5.3. All auxiliary partitions are
empty, and ideal intersections give unions of the eligible tableaux. □
The sum formula specializes as well
where is dominance join. It is the equal-size case of the partition join described
above.
5.2. A simultaneously adapted higher Specht basis.
Dualizing the common harmonic decomposition gives polynomial coordinates adapted
to all the ideals at once. The required change from the original higher Specht polynomials
is triangular.
Corollary 5.7 (A basis adapted to every fixed- ideal).Fix . There are homogeneous polynomials , indexed by and , obtained from by the
same triangular change of multiplicity coordinates for every . For every allowed , the
eligible polynomials form a higher Specht basis of , while every polynomial with an
ineligible index lies in . The change of basis is defined over . All these statements
hold over that ring and over every field of characteristic zero or greater than , with
Specht lattices in place of irreducible modules over the localized ring.
Proof.Fix a shape , total degree, and the chosen tableau . Use the canonical witnesses
of this section in the divided-power form of Section 8. Let be their pairing matrix
with the polynomials . This integer matrix is upper triangular; its diagonal entries
are the products of the two group orders appearing in the divided-power pairing.
They are units in . Put , and, for indices in this shape and degree, define
This is the asserted triangular change of multiplicity coordinates.
To check the vanishing statement first over , let be the hook product and project
the th divided-power witness by after contracting its rejection monomial. Its orbit
is the harmonic copy already constructed, up to a nonzero scalar. The pairing with
the fixed- modified vectors is the identity matrix. The invariant pairing between two
Specht copies of shape is determined by its value on these primitive, adjoint lines.
Consequently the entire copy generated by is orthogonal to whenever . Theorem 5.2
now shows that an ineligible copy is orthogonal to the full harmonic space of the
quotient, so it lies in the ideal. The eligible copies pair nondegenerately with its
harmonic direct sum and therefore give a basis.
For the integral statement, the master quotient has the original family as an
-basis; the same divided-power argument and the word count in Theorem 5.2 prove
this exactly as in Corollary 8.1. The change is invertible over . Each Griffin quotient
is free over by Lemma 6.4, so a polynomial that vanishes after extension to already
vanishes over . Thus all ineligible modified vectors vanish integrally. Their eligible
complements span each quotient because the complete modified family spans the
master quotient. Their number is the rank of the quotient, and independence follows
over , hence over . This proves the integral basis statement and all its reductions
to fields of characteristic greater than . The Specht action is preserved because is
independent of . □
Remark 5.8 (Why the exponent bound is fixed).The common harmonic copies
above are constructed for fixed . They cannot be chosen with the same index labels
simultaneously for all . For example, take , let have bottom row and top row , and
take . This degree-three, shape- index is eligible both for the squarefree quotient with
and for the ordinary coinvariant quotient with . Yet their degree-three harmonic
spaces have zero intersection. Indeed a squarefree cubic has the form . The equation
gives for every pair , and hence over . The coinvariant harmonic space satisfies
this equation because its ideal contains . Thus a nonzero common copy for this index
cannot exist across these two values of .
6. Untruncated ideals and higher Specht free bases
Once the power relations are removed, the auxiliary partition has no rectangle bound, so
its elementary-symmetric factors range over a polynomial parameter ring. The same higher
Specht polynomials therefore form a finite free basis, and the finite-dimensional basis
theorem becomes a description of a flat family.
For , including , remove the power generators and define
Thus , while when . These are Griffin’s ideals and rings from [8, (1.32) and
Section 6]; the notation here distinguishes them from the finite- rings. These quotients are
also the diagonal scheme intersections of Eisenbud–Saltman rank varieties identified in [8,
Corollary 6.4]. Put and
Theorem 6.1 (An explicit higher Specht free basis).The images of are algebraically independent in , and this ring is a finite free module
over . A homogeneous free basis is
Its rank is , with the empty product equal to one. Consequently is Cohen–Macaulay
of dimension . Its graded Frobenius series is
Proof.Fix a degree and choose . The degree- components of and agree. For a
candidate of degree , the inequality
automatically satisfies the finite- bound. Theorem 4.1, together with the
empty-partition case in Theorem 5.2, therefore gives a vector-space basis of
consisting of
with no bound on the largest part of . The map is a bijection to . Thus the
are exactly the monomials in . The single-row tableau is always eligible and has , so
the independence of its multiples also proves algebraic independence of the images
of the . The displayed vector-space basis is precisely the asserted free-module basis
expanded in these monomials.
For the rank, apply RSK to permutations of . Restriction of the insertion tableau
to is the insertion tableau of the subword on those values. Each permutation of
occurs as that subword in permutations of . The number of eligible permutations
of is . This gives the claimed rank; for the same calculation uses the unique
empty permutation. The invariant variables have degrees , and multiplication by
them preserves every Specht copy, giving the Frobenius series. Finally, finite freeness
over the polynomial subalgebra gives dimension and the regular sequence , proving
Cohen–Macaulayness. □
Remark 6.2 (Coefficients of the higher Specht free basis).By the same degreewise
stabilization and Corollary 8.1, the displayed higher Specht free basis works over
every field of characteristic zero or characteristic , and over . The exact unnormalized
family cannot be a basis in characteristic , since its one-row member is . The
underlying algebra has a stronger integral freeness property, proved below using a
different basis.
6.1. A flat lifting of a Springer ring.
Put , the partition obtained by adjoining parts equal to one.
Proposition 6.3 (The special fiber).Over , and hence over every coefficient field,
Moreover
Thus the higher Specht free basis in Theorem 6.1is an explicit flat lifting of the
higher Specht basis of , and
Proof.We have . For every proper subset , the tail sum defining its elementary
generators excludes the first entry, so the generators for and agree. For , their listed
generators are respectively and . This proves the ideal identity over .
Write . The restriction lemma shows that the left condition is equivalent to for
every . Among partitions of , these inequalities are equivalent to . Indeed for the
required inequalities are identical. Beyond that point, either has fewer than nonzero
parts, so its prefix sum is , or each additional part contributes at least one. In the
latter case its prefix sum is at least , which is the required prefix sum of until the total
is reached. The converse is immediate. The asserted lifting and Frobenius identity
now follow from the free-module theorem and the displayed ideal identity. □
The parameter decomposition also holds integrally. We first record why the monomial
quotients used to lift its special fiber are free. Write for the ideal with the same
elementary and power generators over , with meaning that the power generators are
omitted.
Lemma 6.4 (Integral standard monomials).The quotient by is a free abelian
group with Griffin’s standard monomials as basis. In particular, its Hilbert function
is unchanged by extension to any field.
Proof.For the empty partition the ideal is , or when , and the assertion is
immediate. For nonempty partitions, Yu constructs a monic Gröbner basis over
whose polynomials lie in the ideal generated over by the specified generators;
every recursive lift is an integer combination of those elementary generators; see [6,
Section 4, Lemma 4.2 and Theorem 4.3]. For finite , append , as in [6, Lemma 2.9].
Division by these monic polynomials works over . Every original generator has
remainder zero; its integral remainder is also its zero remainder over . Thus this is a
generating set over as well. The same division shows that the standard monomials
span the integral quotient. An integral relation between them would give a relation
over , so they are independent. Finally, tensoring this free quotient with a field gives
the quotient by the same generators. □
Corollary 6.5 (Integral freeness and all coefficient fields).Let , and choose any
homogeneous integral monomial basis of the Springer quotient , where . The same
monomials, lifted to , form a free basis over . In particular this algebra is free of rank
over that polynomial subalgebra. After base change to any field, is Cohen–Macaulay
of dimension , in every characteristic.
Proof.The existence of follows from the integral standard monomials in Lemma 6.4.
Set , with , and map to . The special-fiber identity gives a graded -linear surjection
For completeness, surjectivity follows by induction on degree; reduce a
homogeneous class modulo to an integral combination of the basis monomials, and
express its remainder as , where each has smaller degree.
After tensoring with , source and target have the same Hilbert series by
Proposition 6.3 and Theorem 6.1. The rationalized surjection is therefore an
isomorphism. Its integral kernel is a submodule of a free abelian group, hence
torsion-free, and its rationalization is zero; it must itself be zero. This proves integral
freeness. Tensoring the isomorphism with any field gives the same free-module
decomposition there, and the polynomial parameters remain a regular sequence of
length . □
6.2. A locally finite simultaneous harmonic decomposition.
The untruncated witnesses can also be chosen simultaneously. Their outside exponents
now follow the boosted tableau itself, since no power bound has to be respected;
orthogonal complements are taken degree by degree, where every space is finite
dimensional.
For every standard tableau and every partition with , let be the boosted tableau and
put
Use the column-reading witness on this prefix, and write for its labels in the fixed
tableau . Define
Thus the residue is the rejection count on the prefix and zero outside it. Set
The empty-prefix convention is .
Theorem 6.6 (Simultaneous untruncated harmonic copies).The polynomials generate irreducible Specht copies of shape in degree . They give one locally finite
direct sum decomposition of the polynomial ring such that, for every ,
In particular the case is the whole polynomial ring.
Proof.The elementary-annihilation part of Lemma 2.1 does not require the outside
exponents to be equal; each outside variable contributes at most one to the degree
in of . Its proof therefore shows that annihilates . If and , then and the restriction
lemma gives
Hence the associated belongs to the desired harmonic space.
Lemma 3.1 applies; outside labels are fixed singleton blocks, now with zero
residue, while all retained labels have zero boost and the same column-reading
monotonicities. Consequently the pairing with the polynomials is upper triangular
with positive diagonal in every fixed shape and degree. The primitive-idempotent
argument gives a direct sum of the irreducible copies . There are only finitely many
pairs in each degree. Theorem 6.1 supplies the matching dimension in each degree,
first for and then for every other . This proves the stated decompositions. □
Corollary 6.7 (Untruncated sums and intersections).For any finite expression of
sums and intersections of the ideals , the quotient has a higher Specht vector-space
basis indexed by the corresponding unions and intersections of the eligible pairs ;
ideal intersections give unions, and ideal sums give intersections. Its graded Frobenius
series is the sum of over that index set. This lattice of ideals is distributive.
If all partitions in the expression have the same size , the quotient is finite free over
. A free basis consists of the indexed by the corresponding unions and intersections
of the sets . It is Cohen–Macaulay of dimension .
Proof.Use the common harmonic decomposition degree by degree, as in
Corollary 5.3. All the relevant vector spaces within a fixed degree are finite
dimensional, so orthogonal complements exchange sums and intersections. The same
principal triangular submatrices prove the quotient-basis assertion. For partitions
of a common size , every eligible set has the same unrestricted multiplier condition
. Unions and intersections preserve this product structure, and the free-module
argument of Theorem 6.1 applies directly. The single-row tableau remains eligible
for every expression, so these quotients are nonzero. □
7. Fibers and primary components
The free parameters determine a characteristic polynomial whose distinct roots separate
the coordinates into groups. The rank conditions remain on the zero group, while
translation by a nonzero root leaves an ordinary coinvariant ring on its group. Working
directly in the Artin local factors includes repeated roots and positive characteristic in
the same calculation, and gives both the fiber decomposition and the primary
components.
Let be any field, and let a point of the parameter space have parameter polynomial
where are pairwise distinct and nonzero, for , and . Put and .
Theorem 7.1 (Every fiber).The fiber algebra is
We use . The factors are indexed by assigning coordinate positions to zero and
positions to each . In particular, distinct nonzero roots give , whereas the fiber at
the origin is . Over a field in which does not split, the displayed formula holds after
extension to a splitting field.
Proof.The fiber algebra is finite-dimensional by Corollary 6.5. Its global
characteristic-polynomial identity is
Consequently every closed point assigns the specified roots to coordinate positions
with the stated multiplicities. These points are -rational because splits over . Fix
one closed point, with zero group of size and nonzero groups of sizes , and work in
its Artin local factor.
The monic group polynomials satisfy
Here is a direct justification over an arbitrary characteristic. Modulo the nilpotent
maximal ideal these identities hold, and the proposed factors on the right are
pairwise coprime; a factor of degree zero may simply be omitted. Suppose the
identities hold modulo its th power. Write the errors modulo its st power as , each of
degree smaller than its monic factor . The product identity gives . Reducing
modulo and using coprimality shows ; its degree is smaller than . Induction
through the nilpotent powers proves (7.1). Thus the zero group has its global
elementary polynomials equal to zero, and each shifted group has the same
property.
We check the remaining ideal on the zero group exactly. Write , , and . An original
generator on , where , , and , has degree
Its expansion is
If , then , so and
Each factor therefore lies in . If , then , so it belongs to the global elementary ideal on .
This proves that every original generator vanishes in
where the variables of the latter factors are shifted by . The global fiber equations also
hold in by (7.1).
Conversely, for any proper , a Tanisaki generator satisfies . Hence the original ideal
contains . Its expansion has the form
The coefficient is a unit in the local ring, since each of these coordinates has nonzero
residue. Descending induction on , starting at , therefore recovers every such Tanisaki
generator. The global generators on and the elementary generators in each shifted nonzero
group already follow from (7.1). Thus the local ideal is exactly the ideal defining
.
The algebra is supported at the chosen assignment, so the two ideal inclusions give the
asserted local isomorphism and also show that every assignment occurs. There are
assignments. The decomposition of the finite algebra into its Artin local factors proves the
formula. □
Proposition 7.2 (Trivialization over ordered distinct roots).Let
and map to . Then there is an -equivariant -algebra isomorphism
In the factor indexed by , the coordinate equals , and the remaining coordinates
carry their Springer algebra. Thus the underlying -module over is
where permutes the remaining coordinates. The same formulas hold after base
change to every coefficient field. For , the inverted product is one and the formula is
simply the identity on .
Proof.For each injection, impose and the Tanisaki ideal on the complementary
coordinates. The elementary expansion in the proof of Theorem 7.1 shows integrally
that every generator of vanishes in this quotient. Its global elementary functions are
exactly the , so these assignments define a map from the displayed base change to
the product.
The assignment ideals in are pairwise comaximal. Two distinct assignments either
put different constants in one coordinate, producing the unit , or put a constant in
a coordinate which is nilpotent in the other assignment. In the latter case the other
ideal contains , so their sum contains the unit . When , only the first case is needed.
The Chinese remainder theorem therefore makes the map surjective.
Both source and target are finite free -modules of rank . A surjection between free
modules of the same finite rank is an isomorphism. Variable permutations permute
the injections transitively, and the stabilizer of an injection is the symmetric group on
its complementary positions. This proves the equivariance and the induced-module
description. The empty partition has and scalar factors, so is included in the
argument. □
Proposition 7.3 (Primary components and their multiplicities).Over every field
, the untruncated ideal has the primary decomposition
The same identity holds over , with the displayed terms again primary. Over a
field, its reduced support is the union of the coordinate -planes. Each such plane has
generic multiplicity , and the standard graded multiplicity of the whole quotient is
For , the intersection has its single term .
Proof.Work over a coefficient field or over , writing for the coefficient ring and . For
, put and . Expanding an elementary generator on , with and , shows that ; each
elementary factor on has degree greater than . For , there is only one component
and the ideal identity is immediate. We prove the identity for next.
The quotient is . Its global elementary parameters equal the elementary functions
of the outside variables, because all positive global elementary functions on vanish.
The polynomial ring on those variables is free of rank over its elementary-symmetric
subalgebra, by the empty-partition case of integral freeness and base change.
Therefore is free over , with acting by , of rank .
In , the parameter equals . After inverting , every outside coordinate is a unit
and every coordinate on is nilpotent. The ideals consequently become pairwise
comaximal; for two distinct -subsets, one coordinate is nilpotent in one quotient and
a unit in the other. The Chinese remainder theorem gives a surjection
Both modules are free over of rank , so this map is an isomorphism. Since is
-free, multiplication by is injective. Thus the kernel before localization is already
zero, proving . This also includes , when there is only the component .
For the primary assertion, valid also when , put . The quotient by is graded by
degree in , with each graded piece free over the domain . An element outside has a
nonzero degree-zero residue, which acts injectively on the lowest nonzero graded piece
of any element. Every element of is nilpotent in the quotient. Thus is -primary. The
primes are distinct and incomparable, so the primary decomposition is irredundant
and these are exactly its minimal primes.
Over a field, the localization at has length . There are coordinate components,
each of degree one, giving standard graded multiplicity . □
Corollary 7.4 (Reducedness and the étale locus).Over every field , the algebra is reduced if and only if has at most one part, allowing the empty partition.
For a parameter fiber, factor over an algebraic closure as in Theorem 7.1.
That fiber is geometrically reduced if and only if has at most one part and every
nonzero-root multiplicity equals one. Consequently the finite flat map to is étale
over the squarefree parameter polynomials when , and over the squarefree with when with . Its étale locus is empty when . For , the constant polynomial satisfies
the squarefree and nonzero-constant conditions by convention. Ordinary reducedness
of a nonsplit fiber over an imperfect field is not asserted by this criterion.
Proof.If has one part, the primary components in Proposition 7.3 are the coordinate
prime ideals themselves, since . Their intersection is radical; the empty-partition case
is the polynomial ring. If has more than one part, the localization along each minimal
component has length and residue field . It is a nonreduced Artin local algebra, so cannot be reduced.
After algebraic closure, each local factor in the all-fiber formula is a tensor product
of Artin local algebras with the same residue field. It is reduced exactly when its
dimension is one. The Springer factor has dimension one exactly when its partition
has at most one part; a nonzero-root factor has dimension , which equals one exactly
when . These are dimensions as integers, and the conclusion is unchanged in small
characteristic. This proves the geometric-reducedness criterion.
A finite flat morphism is étale at a point exactly when its local geometric fiber
is reduced. Substituting in the criterion gives the three listed cases. If , the map is
the single Springer algebra over ; it is étale exactly when has at most one part,
consistently with the stated convention. □
7.1. Regularity and the canonical module. The regular sequence of elementary
parameters transfers graded invariants from the Springer fiber to the untruncated ring. Its
top degree determines regularity, and concentration of its socle in that degree determines
the canonical module.
Corollary 7.5 (Regularity).For and , over every field one has
Proof.Put . The top degree of is . This also follows directly from our Springer basis;
for an eligible , its column witness has , so
The last inequality is the usual reversal of under dominance, obtained by
summing the row-prefix inequalities. The row-superstandard tableau of shape has
catabolizability type and cocharge , so equality is attained. Integral freeness makes
the Hilbert series unchanged in every characteristic.
Write the standard graded Hilbert series of as . The flat-lifting identity gives
The quotient is Cohen–Macaulay over every field, so its regularity is the degree
of this Hilbert numerator. Therefore
□
Corollary 7.6 (Levelness, type and the Gorenstein cases).Let , and let be the
number of standard tableaux of shape . Over every coefficient field, is level of
Cohen–Macaulay type ; its canonical module is generated in the single degree
It is Gorenstein exactly when is empty, a single row, or a single column.
Equivalently, is a column, including the empty partition, or and . These are also
exactly the complete-intersection cases.
Proof.We first verify the necessary Springer socle statement in every characteristic.
The empty partition gives the field itself. For , put . Use divided-power symbols ,
with monomial contraction when , and zero otherwise. For a filling of shape , define
These degree- alternants annihilate integrally. Indeed over the divided-power
identification sends to ; the direct annihilation lemma therefore proves the
contraction identities over , and their integer coefficients prove them over and every
field.
Let be the nonnegative vectors bounded coordinatewise by a shuffle of the column
staircases
At exponent bound , the Griffin quotient is integrally. Indeed, modulo the global
elementary generators, , and the proper-subset threshold makes zero. When , that
polynomial is identically zero. Thus over . Griffin’s monomial-basis theorem gives ;
see [7, Section 3.2 and Theorem 4]. For , choose such a shuffle and place its indices in
the corresponding columns of a filling . The column alternant then has lexicographic
leading exponent and coefficient .
Contract by . Its leading term is with coefficient ; every other surviving
exponent remains lexicographically smaller after the same subtraction. The -indexed
contractions are consequently linearly independent over every field. The integral
standard monomials give the same dimension in all characteristics, so these
contractions form the entire divided-power inverse system of . It is therefore generated
under contraction in the single degree . Dualizing the multiplication maps shows that
the socle of is concentrated in degree .
Its dimension is . Over characteristic zero, the top space of alternants is the usual
Specht space; the map from row tabloids to monomials with their row exponents
is injective and sends polytabloids to the column alternants. Its dimension is . The
characteristic-independent Hilbert function then gives the same top dimension over
every field. Thus every is level of type .
Now use the regular sequence in , where , whose quotient is . For canonical
modules, regular reduction gives
Graded Nakayama therefore transfers generation in one degree and the number of
generators from to . The former is generated in degree by elements. This proves the
asserted degree and type, with no restriction on the characteristic. Finally exactly
for the empty partition, a row, or a column, which gives the Gorenstein criterion. The
empty-partition case has and recovers the polynomial ring; when , the statement is
exactly the Springer socle statement.
For , the proper-subset generators vanish identically, so . This is a regular
sequence; the polynomial ring is free over its full elementary-symmetric polynomial
subalgebra. For and , the ideal is . Thus every listed case is a complete intersection.
Conversely, every complete intersection is Gorenstein, so there are no others. □
8. The exact characteristic range
Since the integral standard-monomial basis in Lemma 6.4 fixes the dimension in every
characteristic, the coefficient problem is entirely one of independence for the specified
higher Specht polynomials. The coefficient pairing removes the factorials introduced by
ordinary differentiation, including those depending on the exponent bound; only orders of
subgroups of remain.
Corollary 8.1 (The exact characteristic range).Over a field,
the specified unnormalized Springer and -Springer higher Specht families, with the
prescribed factors, are bases if and only if the characteristic is zero or is a prime .
In these characteristics each fixed-Springer family, and each fixed--Springer family,
is an irreducible Specht module of its stated shape. There is no restriction involving
.
Moreover, the displayed polynomials form bases of the corresponding quotients
over in both cases.
Proof.Use the coefficient functionals of Lemma 3.1. They are defined over .
Equivalently, in the divided-power module with contraction when and zero
otherwise, they pair against
and extract the coefficient of . Over , the identification takes to . The annihilation
identities therefore hold over , hence over because their coefficients are integral. The
functionals descend after any field base change.
Their triangular diagonals are already given by (3.1)
Both group orders divide . Thus every diagonal is a unit after inverting , with no
condition on the outside exponent .
For , the symmetric-group algebra is split semisimple, and the usual
standard-polytabloid Specht modules are irreducible. The equivariant argument that
promotes fixed- independence to independence of all the Specht copies therefore
applies unchanged. One may equivalently extend first to an algebraic closure and
then descend linear independence. Lemma 6.4 gives the required dimension in
every characteristic, so the independent families are bases. Conversely, the one-row
tableaux and the empty auxiliary partition always occur among the candidates.
Their polynomial is , which is zero in every characteristic . The specified family
therefore cannot be a basis in any such characteristic. The quotients here are nonzero;
implies , so every defining elementary generator has positive degree, as do the power
generators.
For the last assertion, express the proposed integral polynomials in the integral
standard-monomial basis of Lemma 6.4. Their square coefficient matrix has a
nonzero integer determinant. The field assertion shows that no prime larger than
divides this determinant. It is therefore a unit in , proving the claimed integral bases.
□
References
[1]
R. Chou, M. Gillespie, and M. Hanada, Nonvanishing higher Specht polynomials and a
construction for three row and hook shape Garsia–Procesi modules, arXiv 2609.12255v1, 2026.
[2]
R. Chou and M. Hanada, Representation theoretic bases for the -Springer module, arXiv
2509.24252v1, 2025.
[3]
J. Blasiak, An insertion algorithm for catabolizability, European Journal of Combinatorics 33
(2012), 267–276. DOI 10.1016/j.ejc.2011.09.042.
[4]
A. M. Garsia and C. Procesi, On certain graded -modules and the -Kostka polynomials,
Advances in Mathematics 94 (1992), 82–138.
[5]
S. Ariki, T. Terasoma, and H.-F. Yamada, Higher Specht polynomials, Hiroshima Mathematical
Journal 27 (1997), 177–188.
[6]
T. Yu, Constructing a Gröbner basis of Griffin’s ideal, Mathematische Zeitschrift 309 (2025),
article 9. DOI 10.1007/s00209-024-03621-6.
[7]
B. Rhoades, T. Yu, and Z. Zhao, Harmonic bases for generalized coinvariant algebras,
Electronic Journal of Combinatorics 27(4) (2020), P4.16. DOI 10.37236/9610.
[8]
S. T. Griffin, Ordered set partitions, Garsia–Procesi modules, and rank varieties, Transactions
of the American Mathematical Society 374 (2021), 2609–2660. arXiv 2004.00788v2.