Cocharge and Skewing Formulas for Δ-Springer Modules and the Delta Conjecture (2024)

Abstract

1 Introduction and Main Results

Schur positivity is a central focus of algebraic combinatorics. One famous example is the Macdonald Positivity Conjecture, proven by Haiman [20], which states that the symmetric Macdonald polynomials |$\widetilde{H}_{\mu }(x;q,t)$| expand in the Schur basis with positive coefficients in |$\mathbb{Z}_{+}[q,t]$|⁠. The proof uses the geometry of the Hilbert scheme |$\text{Hilb}_{n}({\mathbb{C}}^{2})$| of arrangements of |$n$| points in the plane |${\mathbb{C}}^{2}$|⁠, and no direct combinatorial proof or explicit formula is yet known.

The Delta Conjecture [17], which generalizes the recently-proven Shuffle Theorem [6], motivates a major current area of research in symmetric function theory (e.g., [2], [18],[19], [25]). It states two combinatorial formulas, in terms of parking functions, for |$\Delta ^{\prime}_{e_{k-1}}e_{n}$| where |$\Delta ^{\prime}_{f}$| is a particular eigenoperator of the Macdonald polynomials defined for any symmetric function |$f$|⁠. One of the two Delta Conjecture formulas has been proven in [2, 7].

The Shuffle Theorem concerns the special case when |$k=n$|⁠, in which |$\Delta ^{\prime}_{e_{n-1}} e_{n}$| is the bi-graded Frobenius series (in |$q,t$|⁠) of the diagonal coinvariant ring

where |$I_{n}$| is generated by the |$S_{n}$|-invariants with no constant term under the diagonal action of |$S_{n}$| permuting the |$x$|’s and |$y$|’s simultaneously. While the Shuffle Theorem gives a monomial expansion for |$\Delta ^{\prime}_{e_{n-1}}e_{n}$|⁠, an explicit formula for the Schur expansion is not known (and similarly for the Delta Conjecture).

In particular, the decomposition of a graded |$S_{n}$|-module |$R=\bigoplus _{d} R_{d}$| into irreducibles can be described by its graded Frobenius character

where |$R_{d}$| is the |$d$|-th graded piece and |$\text{Frob}$| is the additive map on representations that sends the irreducible |$S_{n}$|-module |$V_{\nu }$| to the Schur function |$s_{\nu }$|⁠. For a bi-graded module, we use two parameters |$q,t$| and obtain a bi-variate generating series. This means that determining the Schur expansion for Macdonald polynomials, the Shuffle theorem polynomials, or those of the Delta Conjecture would lead to a deeper understanding of the |$S_{n}$|-representation theory of the associated (bi-)graded modules.

In the one-parameter case, setting |$t=0$| often leads to more tractable problems. For instance, a famous result of Lascoux and Schützenberger was their discovery of the cocharge statistic on Young tableaux to give a combinatorial formula for the Schur expansion of the (modified) Hall-Littlewood polynomials |$\widetilde{H}_{\mu }(x;q)$|⁠, which are the |$t=0$| specialization of the Macdonald polynomials. The polynomials |$\widetilde{H}_{\mu }(x;q)$| are the graded Frobenius character of the Garsia-Procesi modules |$R_{\mu }$|⁠. These |$S_{n}$|-modules in turn are the cohom*ology rings of Springer fibers |$\mathscr{B}_{\mu }$|⁠. The cocharge statistic therefore resolved the natural question of how |$R_{\mu }$| decomposes into irreducible |$S_{n}$|-modules.

In particular, for a partition |$\mu $|⁠, define |$\text{SSYT}(\mu )$| to be the set of all (straight shape) semistandard Young tableaux of content |$\mu $|⁠, meaning that the tableau entries consist of |$\mu _{i}$| copies of |$i$| for each |$i$|⁠, and the entries are weakly increasing across rows and strictly increasing up columns in French notation (as in the “device” part of the tableau at left in Figure 1). Lascoux and Schützenberger showed that

where |$\text{sh}(T)$| is the shape of the tableau |$T$|⁠, that is, the partition whose |$i$|-th part is the length of the |$i$|-th row of |$T$| from the bottom, and |$s_{\text{sh}(T)}$| is the corresponding Schur function. Above, |$\widetilde{K}_{\nu ,\mu }(q)$| is the |$q$|-Kostka polynomial, and |$\text{cc}$| is the cocharge statistic as defined in Section 2.

One of the main results of this article generalizes the Lascoux–Schützenberger formula to the cohom*ology rings of the |$\Delta $|-Springer varieties, which were recently introduced by Levinson, Woo and the second author [15]. These graded |$S_{n}$|-modules are denoted by |$R_{n,\lambda ,s}$| and simultaneously generalize both the Garsia-Procesi modules |$R_{\mu }$| and the generalized coinvariant rings |$R_{n,k}$| that were defined by Haglund, Rhoades, and Shimozono [18] to give an algebraic realization of the Delta Conjecture polynomial |$\Delta ^{\prime}_{e_{k-1}}e_{n}$| at |$t=0$|⁠. We obtain this result by connecting the |$\Delta $|-Springer varieties to the theory of partial resolutions of nilpotent varieties due to Borho and MacPherson [3].

The rings |$R_{n,\lambda ,s}$|⁠, first introduced in [14], are defined for integers |$n,s$| and a partition |$\lambda $| with |$|\lambda |=k\le n$| and |$s\ge \ell (\lambda )$|⁠. In the special case when |$n=|\mu |$|⁠, the ring |$R_{n,\mu ,s}$| coincides with |$R_{\mu }$|⁠. When |$\lambda = (1^{k})$| and |$s=k$|⁠, the ring |$R_{n,\lambda ,s}$| coincides with |$R_{n,k}$|⁠. Because the common generalization |$R_{n,\lambda ,s}$| has a geometric interpretation as the cohom*ology rings of the |$\Delta $|-Springer varieties |$Y_{n,\lambda ,s}$| [15], we refer to them here as the |$\Delta $|-Springer modules.

1.1 New skewing, charge, and cocharge formulas for R_n,λ ,s

We prove that the graded Frobenius character |$\widetilde{H}_{n,\lambda ,s}:=\text{grFrob}(R_{n,\lambda ,s})$| has the following skewing formula.

In the above statement, |$s_{\nu }^{\perp }$| denotes the adjoint operator to multiplication by |$s_{\nu }$| with respect to the Hall inner product on symmetric functions.

The proof of Theorem 1 relies heavily on the work of Borho and MacPherson on partial resolutions of the nilpotent cone. We show that the |$\Delta $|-Springer varieties |$Y_{n,\lambda ,s}$| are instances of the family of varieties studied in their work [3]. We prove a rational smoothness condition that enables us to use a result in [3] derived using the theory of perverse sheaves to obtain the Frobenius character.

As an immediate corollary, we have the following simple formula for the symmetric function in Delta Conjecture at |$t=0$|⁠. We write |$\text{rev}_{q}$| for the operation of reversing the coefficients of the |$q$| polynomial, by setting |$q\to q^{-1}$| and multiplying by |$q^{d}$| where |$d$| is the degree. We also write |$H_{\mu }(x;q) = \text{rev}_{q}(\widetilde{H}_{\mu }(x;q))$| for the (transformed) Hall-Littlewood symmetric functions.

We now provide a combinatorial Schur expansion for |$\text{grFrob}(R_{n,\lambda ,s})$| that generalizes Equation (1). We first make more rigorous the definition of the partition |$\Lambda $| mentioned above.

We prove the following formula for the graded Frobenius character of |$R_{n,\lambda ,s}$|⁠, which was originally conjectured in [10].

We think of the battery as storing extra charge for the device. The |$q$|-exponent |$\binom{s-1}{2}(n-k)$| is the largest amount of cocharge that may be stored in the battery.

In order to prove Theorem 3 from Theorem 1, we apply the operator |$s_{((n-k)^{s-1})}^{\perp }$| directly to Equation (1), and in the process, we also obtain the following formula (in the Delta Conjecture case) in terms of Littlewood-Richardson coefficients and |$q$|-Kostka polynomials.

By applying |$\text{rev}_{q}$| to Theorem 3, we can obtain the following alternative simpler expansion in terms of the generalized charge statistic.

Specializing to the case relevant to the Delta Conjecture, |$\lambda =(1^{k})$| and |$s=k$|⁠, we have a new Schur expansion for the expression in the Delta Conjecture at |$t=0$|⁠.

Since the proof of Theorems 3 and 4 that we present here is essentially geometric in nature, it is also of interest to find a more direct combinatorial proof, using the existing expansions of |$\text{grFrob}(R_{n,\lambda ,s})$| in terms of monomials or sums of Hall-Littlewood polynomials. The following theorem summarizes some of our progress towards a combinatorial proof.

This proposition was stated without full proof details in the conference proceedings article [10], and we provide the complete proofs in this paper. In the companion paper [11] to this work, the authors will provide combinatorial proofs of two additional special cases using a new formula in terms of creation operators and the Loehr-Warrington algorithms on abaci.

1.2 Outline

After establishing background definitions and notation in Section 2, we prove Theorem 1 in Section 3. We then prove Theorem 3 and Theorem 4 in Section 4 and check that the highest degree terms agree with what we would expect. In Section 5, we give a combinatorial proof of Theorem 4 at |$s=2$|⁠, and in Section 6, we prove it for the |$s_{(n)}$| coefficient in the Delta conjecture case. In Section 7, we give conjectural formulas for the Delta Conjecture symmetric function for |$t$| degree at most |$2$| in terms of skewing sums of Hall-Littlewood polynomials. Finally, in Section 8, we outline potential future research directions.

2 Background

We now recall some background and definitions on tableaux operations, cocharge and charge, and geometry related to the |$\Delta $|-Springer varieties. We refer to [8] for the definition of the basic operation of jeu de taquin rectification on skew semistandard Young tableaux.

2.1 Tableaux and insertion

We write partitions |$\lambda =(\lambda _{1},\ldots ,\lambda _{r})$| with their parts nonincreasing: |$\lambda _{1}\ge \lambda _{2}\ge \cdots \ge \lambda _{r}$| and write |$r=\ell (\lambda )$| for the length of |$\lambda $|⁠. We draw them in French notation, with |$\lambda _{i}$| boxes in the |$i$|-th row from the bottom, and use the shorthand |$(a^{b})=(a,a,a,\ldots ,a)$| to denote the |$b\times a$| rectangular partition with |$b$| parts of size |$a$|⁠. A semistandard Young tableau (SSYT) of shape |$\lambda $| is a filling of the boxes of |$\lambda $| that weakly increases across rows and strictly increases up columns. As stated in the introduction, we write |$\text{SSYT}(\mu )$| for the set of semistandard Young tableaux of content |$\mu $| (and any shape).

The reading word of a tableau is the word formed by concatenating the rows from top to bottom. For instance, the reading word of the battery-powered tableau in Figure 1 is

The RSK insertion or row bumping of a letter |$i$| into a tableau |$T$| is the tableau |$T^{\prime}$| formed by inserting |$i$| into the bottom row |$R_{1}$| of |$T$|⁠, where it is placed at the end if |$i$| is greater than or equal to every element of |$R_{1}$| and otherwise it replaces the leftmost entry |$m$| of |$R_{1}$| that is greater than |$i$|⁠. Then |$m$| is inserted into the second row |$R_{2}$| in the same manner, and so on until the process is complete and a new entry is added. RSK insertion is reversible given the final bumped entry [8], and we call the reverse process unbumping.

We also say the RSK insertion of a tableau |$B$| into a tableau |$D$| (such as in the case of a battery |$B$| and device |$D$|⁠) is the tableau |$T^{\prime}$| formed by inserting the letters of the reading word of |$B$| one at a time into |$D$|⁠. We write |$T^{\prime}=D\cdot B$|⁠. It is well known (see [8]) that |$D\cdot B$| is equal to the jeu de taquin rectification of the skew tableau formed by placing |$B$| down-and-right of |$D$|⁠. We use this equivalence implicitly in this paper.

Two words are Knuth equivalent if their RSK insertions (one letter at a time inserted into the empty tableau from left to right) are equal.

A horizontal strip is a skew shape in which no two boxes appear in the same column. It is known that RSK inserting a nondecreasing sequence into a tableau |$T$| extends the shape of |$T$| by a horizontal strip.

2.2 Symmetric functions

We work in the ring of symmetric functions over |$\mathbb{Q}$| in the countably infinite set of variables |$x_{1},x_{2},x_{3},\ldots $|⁠, which we often simply abbreviate as |$x$|⁠. We refer to [26] for the definitions of the Schur functions |$s_{\lambda }(x)$| and the elementary symmetric functions |$e_{\lambda }(x)$|⁠.

We recall that the Hall inner product is the symmetric inner product |$\langle ,\rangle $| on the space of symmetric functions for which |$\langle s_{\lambda },s_{\mu }\rangle =\delta _{\lambda \mu }$|⁠. We write |$f^{\perp }$| for the adjoint operator to multiplication by |$f$| with respect to the Hall inner product; that is,

It is known that |$s_{\mu }^{\perp } s_{\nu }=s_{\nu /\mu }$|⁠. We now observe a representation theoretic meaning of the operator |$s_{\mu }^{\perp }$| (our statement can essentially be found in different language in [26], and we include details and proof here for completeness). In the below statement, the |$V_{\mu }$|-isotypic component of an |$S_{n}$|-module |$W$| is the sum of all copies of the irreducible Specht module |$V_{\mu }$| in the decomposition of |$W$| into irreducibles.

Also recall the omega involution on symmetric functions, which may be defined as the unique linear operator |$\omega $| such that |$\omega (s_{\lambda }) = s_{\lambda ^{*}}$|⁠, where |$\lambda ^{*}$| is the conjugate partition of |$\lambda $|⁠.

Given a symmetric function |$f(x;q)$| with coefficients in |$\mathbb{Q}[q]$|⁠, we have the |$q$|-reversal operator |$\text{rev}_{q}$| that reverses the coefficients of |$f$| as a polynomial in |$q$|⁠. Precisely, if |$f(x;q)$| has |$q$| degree |$d$| as a polynomial in |$q$| with symmetric function coefficients, then |$\text{rev}_{q}(f(x;q)) = q^{d} f(x;1/q)$|⁠.

2.3 Charge and cocharge

We first define cocharge on words, using the reading word of the tableau |$T$| in Figure 1 as a running example:

The first cocharge subword is formed by searching right to left in the reading word for a |$1$|⁠, then continuing from that position to search for a |$2$| (wrapping around the end cyclically if necessary), and so on until we have reached the largest letter of the word:

The cocharge labeling of a permutation is computed by searching right to left cyclically as before, labeling the entries |$1,2,3,\ldots $| in order, and starting by labeling the |$1$| with a |$0$| and incrementing the label if and only if the next entry is to the left of the previous:

We then similarly find and label the second cocharge subword among the unlabeled letters:

We continue to iterate this process on the unlabeled letters until all have been labeled:

In Figure 1, the cocharge labels on the reading word elements are shown in the corresponding squares at right. The charge labels are placed in the same order as cocharge labels except we increment when the next element is to the right of the previous.

The cocharge (resp. charge) of |$T$|⁠, written |$\text{cc}(T)$| and |$\text{ch}(T),$| respectively, is the sum of the cocharge (resp. charge) labels of its reading word. Therefore, the cocharge of the word above is |$3+2+2+1+1+2+1=12$|⁠.

Cocharge and charge are invariant under bumping: we have |$\text{ch}(D\cdot B)=\text{cc}(T^{\prime})$| and |$\text{ch}(D\cdot B)=\text{ch}(T^{\prime})$| where |$T^{\prime}$| is the insertion of |$B$| into |$D$|⁠. This is because RSK insertion preserves the Knuth equivalence class of the reading word [8], and cocharge and charge are invariant under Knuth equivalence [22].

The maximum possible cocharge of a semistandard Young tableau of a given content |$\nu $| occurs in the unique such tableau that has shape |$\nu $| as well. In this case, the cocharge label of each of the |$\nu _{i}$| entries in the |$i$|-th row is |$i-1$|⁠. This leads to the following definition, which we use frequently throughout.

2.4 Hall-Littlewood polynomials

We recall the Hall-Littlewood polynomials, which are symmetric functions with coefficients in a parameter |$q$|⁠. Given a partition |$\mu $| of |$n$|⁠, the transformed Hall-Littlewood polynomial |$H_{\mu }(x;q)$| is the symmetric function with Schur expansion given by the charge statistic,

Alternatively, applying the |$\text{rev}_{q}$| operator we get the modified Hall-Littlewood polynomial, with Schur expansion given by the cocharge statistic,

As mentioned in the introduction, the modified Hall-Littlewood polynomial |$\widetilde{H}_{\mu }(x;q)$| is the graded Frobenius character of |$R_{\mu }$|⁠, the cohom*ology ring of the Springer fiber |$\mathscr{B}_{\mu }$|⁠, which we define in the next subsection.

2.5 Springer fibers and Δ-Springer varieties

Let |$G = GL_{K}({\mathbb{C}})$|⁠, let |$B$| be the Borel subgroup of invertible upper triangular matrices, and let |$\mathscr{B}(K) = G/B$| be the complete flag variety, which may be identified with the space of complete flags |$\mathscr{B}(K) = \{F_{\bullet } = (F_{1}\subset F_{2}\subset \cdots \subset F_{K})\mid \dim (F_{i}) = i, F_{K} = {\mathbb{C}}^{K}\}$|⁠. In particular, given |$gB\in G/B$|⁠, the corresponding flag is |$g F_{\bullet }^{\text{std}}$| where |$F_{i}^{\text{std}}$| is the flag spanned by the first |$i$| standard basis vectors. We also let |${\mathcal{N}}$| be the nilpotent cone of |$K\times K$| nilpotent matrices.

The group |$G$| acts on |${\mathcal{N}}$| via the adjoint action, |$Ad(g)x := gxg^{-1}$|⁠. For |$x\in{\mathcal{N}}$| nilpotent, we write |$\text{JT}(x)$| for the Jordan type of |$x$|⁠, which is the partition of |$K$| recording the Jordan block sizes of |$x$| in Jordan canonical form. The set of all |$x\in{\mathcal{N}}$| with a fixed Jordan type |$\mu $| is an orbit of |${\mathcal{N}}$| under the adjoint action of |$G$|⁠, which we denote by |${\mathcal{O}}_{\mu }$|⁠.

Given |$x\in{\mathcal{N}}$|⁠, the Springer fiber associated to |$x$| is

The isomorphism type of |$\mathscr{B}_{x}$| only depends on |$\text{JT}(x)$|⁠, and thus we may write |$\mathscr{B}_{\mu }$| for any |$x\in{\mathcal{O}}_{\mu }$|⁠.

Springer discovered that these varieties have the remarkable property that the symmetric group |$S_{K}$| acts on the cohom*ology ring |$H^{*}(\mathscr{B}_{\mu };{\mathbb{Q}})$| and the top nonzero cohom*ology group is an irreducible Specht module,

More generally, Hotta and Springer [21] proved that

In [15], Levinson, Woo, and the second author introduced the |$\Delta $|-Springer varieties that generalize the Springer fibers and give a geometric realization of the symmetric function in the Delta Conjecture at |$t=0$|⁠.

Let |$n,\lambda ,s$| be as in Definition 1.2, and let |$K = |\Lambda | = k + (n-k)s = n+(n-k)(s-1)$|⁠. Let |$x$| be a nilpotent |$K\times K$| matrix with Jordan-type |$\Lambda $|⁠, and let |$P$| be a parabolic subgroup of |$G = GL_{K}$| with block sizes |$\alpha = (1^{n},(n-k)(s-1))$|⁠, so that |$\mathscr{P} = G/P$| corresponds to partial flags |$(F_{1}\subset F_{2}\subset \cdots \subset F_{n} \subset F_{n+1})$| with |$\dim (F_{i}) = i$| for |$i\leq n$| and |$F_{n+1} = {\mathbb{C}}^{K}$|⁠. The |$\Delta $|-Springer varieties are defined to be

Recall that we write |$k=|\lambda |$|⁠. When |$k=n$| (and |$s$| is arbitrary), |$Y_{n,\lambda ,s} \cong \mathscr{B}_{\lambda }$|⁠, so these varieties generalize the Springer fibers.

Levinson, Woo, and the second author proved that the |$\Delta $|-Springer varieties |$Y_{n,\lambda ,s}$| have several geometric and combinatorial properties that generalize those of Springer fibers:

• |$Y_{n,\lambda ,s}$| is equidimensional of dimension |$\textbf{n}(\lambda ) + (n-k)(s-1)$|⁠.

• There is an |$S_{n}$| action on |$H^{*}(Y_{n,\lambda ,s};{\mathbb{Q}})$|⁠.

• The top cohom*ology group is a skew Specht module |$H^{top}(Y_{n,\lambda ,s};{\mathbb{Q}}) \cong V_{\Lambda /((n-k)^{s-1})}$|⁠.

• |$H^{*}(Y_{n,\lambda ,s})$| has a presentation as a quotient of the polynomial ring |${\mathbb{Z}}[x_{1},\dots , x_{n}],$| which coincides with the ring |$R_{n,\lambda ,s}$| introduced in [14]. In the special case |$\lambda = (1^{k})$| and |$s=k$|⁠, the cohom*ology ring coincides with the generalized coinvariant rings of Haglund, Rhoades, and Shimozono, |$H^{*}(Y_{n,(1^{k}),k};{\mathbb{Q}}) = R_{n,k}$|⁠.

Notably, in the special case when |$\lambda = (1^{k})$| and |$s=k$|⁠, then

so |$Y_{n,(1^{k}),k}$| gives a geometric realization of the symmetric function in the Delta Conjecture at |$t=0$| (up to a minor twist).

2.6 Rational smoothness and intersection cohom*ology

Here |$IH^{*}_{x}$| is the middle local intersection cohom*ology, see [12]. See [3] for a proof of the fact that (1) and (2) above are equivalent. We do not define intersection cohom*ology here, but the essential property of local intersection cohom*ology that we need is that for |$x\in{\mathcal{O}}_{\mu }$|⁠,

which is a result due to Lusztig [23]. See also [27] for more details and related results. In particular, (4) reflects the fact that

where |$\preceq $| is dominance order on partitions of the same size, defined by |$\mu \preceq \nu $| if |$\mu _{1} + \cdots +\mu _{i} \leq \nu _{1} + \cdots + \nu _{i}$| for all |$i$| [24].

We will need the next fact, which follows easily from the Relative Künneth Formula for the local cohom*ology of a product space.

2.7 Borho and MacPherson’s partial resolutions

Let |$P$| be a parabolic subgroup, and let |$\mathscr{P} = G/P$| be the corresponding partial flag variety. Let |$L$| be the Levi subgroup associated to |$P$|⁠, let |${\mathcal{N}}_{L}$| be the nilpotent cone of |$L$|⁠, and finally let |${\mathfrak{p}} = {\mathfrak{l}} \oplus{\mathfrak{n}}$| be the Levi decomposition of |${\mathfrak{p}} = \text{Lie}(P)$|⁠, where |${\mathfrak{l}} = \text{Lie}(L)$| and |${\mathfrak{n}}$| is the nilradical of |${\mathfrak{p}}$|⁠.

Explicitly, |$P$| is the set of invertible block upper triangular matrices with block sizes given by some composition |$\alpha $| of |$K$|⁠, |$G/P$| is the variety of partial flags |$(V_{1}\subseteq V_{2}\subseteq \cdots \subseteq V_{\ell })$| of |${\mathbb{C}}^{K}$| with |$\dim (V_{i}/V_{i-1}) = \alpha _{i}$| for all |$i$|⁠, and |$L$| is the subgroup of invertible block diagonal matrices with block sizes given by |$\alpha $|⁠. The Lie algebra |$\mathfrak{n}$| is the set of block strictly upper triangular matrices, |${\mathcal{N}}_{L}$| is the set of nilpotent block diagonal matrices, |${\mathfrak{p}}$| is the set of block upper triangular matrices, and |${\mathfrak{l}}$| is the set of block diagonal matrices, with block sizes given by the parts of |$\alpha $|⁠.

Borho and MacPherson defined the partial resolutions of the nilpotent cone, defined by

where |$\xi (g,x) = \text{Ad}(g)x=gxg^{-1}$|⁠. Here, the |$\times _{P}$| notation denotes that we are taking the quotient of the product space by the right |$P$| action |$(g,x)\cdot p = (gp,\text{Ad}(p^{-1})x)$|⁠. The variety |$\widetilde{{\mathcal{N}}}^{P}$| has the following alternative description in terms of partial flags. We have an isomorphism

where |$\xi $| is the projection onto the second factor. In particular, when |$P=B$| then |${\mathcal{N}}_{L} = 0$| and |${\mathfrak{n}}$| are the strictly-upper triangular matrices, and hence we recover the usual Springer resolution, which we denote by |$\pi : \widetilde{{\mathcal{N}}} = \widetilde{{\mathcal{N}}}^{B}\to \mathcal{N}$|⁠.

Given |$t\in{\mathcal{N}}_{L}$|⁠, write |$L\cdot t$| for the orbit |$\text{Ad}(L)t$|⁠. Let |$y = (1,t)\in \widetilde{{\mathcal{N}}}^{P}$|⁠. The subspaces

partition |$\widetilde{{\mathcal{N}}}^{P}$| as |$t$| varies over a set of representatives of |$\text{Ad}(L)$|-orbits in |${\mathcal{N}}_{L}$|⁠. Since |${\mathcal{O}}_{y}$| is a fiber bundle over |$G/P$| with fiber |$L\cdot t+{\mathfrak{n}}$|⁠, taking the closure we have

which can be seen by taking the closure on each trivializing open subset of |$G/P$|⁠.

The usual Springer resolution |$\pi :\widetilde{{\mathcal{N}}}\to{\mathcal{N}}$| factors through |$\xi $|⁠. Letting |$\xi _{y}$| be the restriction of |$\xi $| to |$\overline{{\mathcal{O}}}_{y}$|⁠, and letting |$\eta :\widetilde{{\mathcal{N}}}\to \widetilde{{\mathcal{N}}}^{P}$| be the induced map from the natural projection |$G/B\to G/P$| (using the description of |$\widetilde{{\mathcal{N}}}^{P}$| in (6)), we have the following commutative diagram:

Given |$x\in{\mathcal{N}}$|⁠, the generalized Springer fiber is |$\mathscr{P}_{x}^{y}:= \xi _{y}^{-1}(x) = \overline{{\mathcal{O}}}_{y}\cap \xi ^{-1}(x)$|⁠. Note that the ordinary Springer fibers |$\mathscr{B}_{x}$| are recovered when |$P=B$| is the full Borel subgroup (and |$y=0$|⁠).

The variety |$\mathscr{P}_{x}^{y}$| can alternatively be described in terms of partial flags as follows. Given |$(g,x^{\prime})\in \mathscr{P}_{x}^{y}$|⁠, let |$F_{\bullet }\in \mathscr{P}$| be the partial flag corresponding to |$gP$|⁠. Since |$(g,x^{\prime})\in \mathscr{P}_{x}^{y}$|⁠, then by definition |$x = \text{Ad}(g)x^{\prime}$|⁠, and it can be checked that |$xF_{i}\subseteq F_{i}$| for all |$i$|⁠. Thus, |$x$| induces a nilpotent endomorphism of |$F_{i}/F_{i-1}$| for all |$i$|⁠, which we denote by |$x|_{F_{i}/F_{i-1}}$|⁠. Letting |$t = t_{1}+\cdots +t_{\ell }$| be the block decomposition of |$t$|⁠, it then follows from (8) that

Let |$\mathscr{B}(L)_{t}$| be the Springer fiber of |$t$| in the flag variety |$\mathscr{B}(L)\cong \mathscr{B}(\alpha _{1})\times \cdots \times \mathscr{B}(\alpha _{\ell })$| for the group |$L$|⁠. In other words,

Borho and MacPherson showed that |$\eta ^{-1}(y)\cong \mathscr{B}(L)_{t}$|⁠. We write |$d_{y} = \dim _{\mathbb{C}}(\eta ^{-1}(y)) = \dim _{\mathbb{C}}(\mathscr{B}(L)_{t})$|⁠.

Let |$\rho (t,1)$| be the irreducible representation of |$W_{L} = S_{\alpha _{1}}\times \cdots \times S_{\alpha _{\ell }}$| on |$H^{top}(\mathscr{B}(L)_{t};{\mathbb{Q}})$|⁠. In other words, |$\rho (t,1) \cong V_{\text{JT}(t_{1})}\otimes \cdots \otimes V_{\text{JT}(t_{\ell })}$| as a |$W_{L}$|-module. Given a |$W$|-module |$V$|⁠, recall that |$V^{\rho (t,1)}$| is the isotypic component corresponding to |$\rho (t,1)$| of the restriction of |$V$| to a |$W_{L}$|-module. Observe that the “partial Weyl group” |$W^{P} = N_{G}(L)/L$| of permutations of the blocks of |$L$| of equal size acts on |$V^{\rho (t,1)}$|⁠.

3 Proof of the Main Theorem

In this section, we prove Theorem 1 using the geometry of Borho–MacPherson partial resolutions. Readers interested in the combinatorial applications of the formula may skip to Section 4.

We begin with a technical lemma that will help us apply Theorem 5 to our setting of |$\Delta $|-Springer varieties.

We now can prove Theorem 1, which we restate here.

As an immediate corollary of Theorem 1, we see how the formula in [15] for the top cohom*ology of |$Y_{n,\lambda ,s}$| follows immediately from the skewing formula.

4 Proofs of the Cocharge and Charge Formulas

In this section, we use Theorem 1 to prove Theorems 3 and 4.

4.1 Proof of Theorem 3

We now deduce the cocharge formula (Theorem 3) from Theorem 1. In particular, we wish to show that

Recall also the following skewing formula for applying an adjoint Schur operator to another Schur function:

We will prove the following more general lemma, from which Equation (13) immediately follows. Define a generalized battery-powered tableau with (not necessarily rectangular) battery shape |$\rho $| and content |$\mu $| to be a pair |$(D,B)$| of semistandard Young tableaux such that |$\text{sh}(B)=\rho $| and the total content of |$D\cup B$| is |$\mu $|⁠. Write |$\mathcal{T}^{+}(\rho ,\mu )$| to be the set of all such pairs |$T=(D,B)$|⁠, and write |$\text{sh}^{+}(T)=\text{sh}(D)$|⁠.

From line (16) above, setting |$\mu =\Lambda _{n,(1^{k}),k}$| and |$\rho =((n-k)^{k-1})$|⁠, we can also deduce

Corollary 1.5 follows immediately by applying the |$\text{rev}_{q}$| operator.

4.2 Proof of Theorem 4

We now deduce the charge version of the main result, Theorem 4, from Theorem 3. For any partition |$\nu $|⁠, recall that |$\textbf{n}(\nu )=\sum _{i} (i-1)\nu _{i}$|⁠.

Dividing out by the factor |$q^{\binom{s-1}{2}(n-k)}$|⁠, we obtain the following corollary.

The value |$d$| matches with the formula given for the top degree of |$\text{grFrob}_{q}(R_{n,\lambda ,s})$| in [14]. In [15], it was shown that the coefficient of |$q^{d}$| is the skew Schur function |$s_{\Lambda /((n-k)^{s-1})}$|⁠. A straightforward application of the Littlewood-Richardson rule shows that this agrees with our formula in Corollary 4.2, and we refer to [10] for details.

Finally, we show that Theorems 3 and 4 are equivalent. Taking the |$q$|-reversal of both sides of Theorem 3, we have

Then the exponent |$\textbf{n}(\lambda )+(n-k)(s-1)-\text{cc}(T)+\choose{s-1}{2}(n-k)$| is equal to |$\textbf{n}(\Lambda )-\text{cc}(T)$|⁠, which is simply |$\text{ch}(T)$| by the definition of charge. This gives Theorem 4.

5 The Case s = 2

In this section, we give a second proof of Theorem 3 in the case when |$s=2$| using combinatorial bijections and previously known formulas for |$\widetilde{H}_{n,\lambda ,s}$|⁠. We start by recalling the Hall-Littlewood expansion of |$\widetilde{H}_{n,\lambda ,s}$|⁠.

5.1 Hall-Littlewood expansion

In [13], it is shown that |$\widetilde{H}_{n,\lambda ,s}$| has the following expansion in terms of Hall-Littlewood polynomials.

where |$\alpha = (\alpha _{1},\dots ,\alpha _{s})\vDash n$| indicates that |$\alpha $| is a weak composition of |$n$| with |$s$| parts, |$\textbf{n}(\mu /\lambda )=\sum _{i} \choose{\mu _{i}^{\prime}-\lambda _{i}^{\prime}}{2},$| and |$\text{coinv}(\alpha )$| is the number of pairs |$i<j$| with |$\alpha _{i}<\alpha _{j}$|⁠.

Note that if |$\alpha $| is a composition such that |$\alpha \supset \lambda $|⁠, then since |$\lambda $| is a partition we also have |$\text{sort}(\alpha )\supset \lambda $|⁠. Thus, we can rearrange the summation above as

where the quantity |$\textbf{n}(\alpha /\lambda )$| above is defined to be |$\textbf{n}(\mu /\lambda )$| where |$\mu =\text{sort}(\alpha )$|⁠.

Substituting (2) into (18) yields

Thus, to prove Theorem 4 it suffices to show that

In particular, it suffices to find a shape-preserving bijection from |$\mathcal{T}^{+}(n,\lambda ,s)$| to

such that, if |$T\in \mathcal{T}^{+}(n,\lambda ,s)$| maps to |$(\alpha ,U)\in \mathcal{A}(n,\lambda ,s)$|⁠, then |$\text{ch}(T)=\text{ch}(U)+\textbf{n}(\alpha /\lambda )+\text{coinv}(\alpha )$|⁠. In the next subsection, we find such a bijection in the case |$s=2$|⁠.

5.2 Combinatorial proof for s = 2

For the remainder of this section, let |$\lambda =(\lambda _{1},\lambda _{2})$| be a partition of size |$k$| with |$\lambda _{1}\ge \lambda _{2}\ge 0$|⁠, and let |$\text{Alpha}(n,\lambda ,2)$| be the set of all (weak) compositions |$\alpha = (\alpha _{1},\alpha _{2})$| of size |$n$| such that |$\alpha \supset \lambda $|⁠.

As a running example, let |$n=11$|⁠, |$\lambda = (3,1)$|⁠, |$s=2$|⁠, and |$\alpha = (5,6)$|⁠. Then |$\textbf{n}(\alpha /\lambda ) + \text{coinv}(\alpha ) = 2+1=3$|⁠. Since |$\text{sort}(\alpha ) = (6,5)$|⁠, then |$\varphi (\alpha ) = (3,8)$|⁠.

We now construct a bijection from |$\mathcal{A}(n,\lambda ,2)$| to |$\mathcal{T}^{+}(n,\lambda ,s)$| as follows.

Continuing our running example with |$\alpha = (5,6)$|⁠, letting |$U$| be the following tableau with |$\text{ch}(U)=2$|⁠, then |$\psi (\alpha , U)$| is as below:

For our running example, we have |$\Lambda _{n,\lambda ,s} = (10,8)$| and

so that |$\text{ch}(\Phi (\alpha ,U)) = 5 = \text{ch}(U) + \textbf{n}(\alpha /\lambda )+\text{coinv}(\alpha )$|⁠.

6 The s_(n) Coefficient in the R_n,k Case

We now consider the setting in which |$\lambda =(1^{k})$| and |$s=k$|⁠, so that |$R_{n,\lambda ,s}=R_{n,k}$| and |$\widetilde{H}_{n,\lambda ,s} = \text{grFrob}(R_{n,k})$|⁠, and give a direct combinatorial proof of Theorem 3 for the coefficient of |$s_{(n)}$| in this setting. We recall the positive Schur expansion of |$\text{grFrob}(R_{n,k})$| given in [1]. An ordered set partition, or OSP, of |$n$| is a partition of |$\{1,2,\ldots ,n\}$| into a disjoint union of subsets called blocks, along with an ordering of the blocks from left to right. For instance, |$(45|367|28|19)$| denotes an OSP of |$9$|⁠.

A descent of a permutation |$\pi $| is an index |$d$| such that |$\pi _{d}>\pi _{d+1}$|⁠, and the major index of |$\pi $| is the sum of its descents. The minimaj of an OSP, first introduced in the context of the Delta conjecture in [17], is the major index of the minimaj word formed by ordering each block’s entries from least to greatest and then reading the letters in the OSP from left to right. For instance, the associated word to |$(45|367|28|19)$| is |$453672819$|⁠, and it has descents in positions |$2,5,7$|⁠, so the minimaj is |$2+5+7=14$|⁠.

The reading word |$\text{rw}(P)$| of an OSP |$P$| (different from its minimaj word) is formed by reading the smallest entry of each block from right to left, and then the remaining entries from left to right. For instance, the reading word of |$(45|367|28|19)$| is |$123456789$|⁠.

It was shown in [18] (using the work of [17]) that there is a more general set of ordered multiset partitions into |$k$| blocks, |$\mathcal{O}\mathcal{P}_{n,k}$|⁠, and a minimaj statistic on them such that

where |$\text{wt}(\pi )$| is the tuple whose |$i$|-th term is the number of |$i$|’s in |$\pi $|⁠. In [1], a crystal structure is given on ordered multiset partitions that is compatible with the minimaj statistic, thereby grouping the terms of the above monomial expansion into a Schur expansion:

where |$\widetilde{e}_{i}$| are the raising operators of the crystal, which we define below.

In particular, the coefficient of |$s_{(n)}$| in the above expansion (taking into account the conjugation via |$\omega $|⁠) is equal to

where |$\text{OSP}(n,k)$| is the set of ordered set partitions with entries |$1,2,\ldots ,n$| and |$k$| blocks.

The crystal raising operators |$\widetilde{e}_{i}$| were defined in [1] via the reading word described above. In particular, |$\widetilde{e}_{i}(P)=0$| if and only if, in the reading word, the number of |$i$|’s is always greater than or equal to the number of |$i+1$|’s as we read the word from left to right. Thus, if |$P$| has content |$(1^{n})$|⁠, we have |$\widetilde{e}_{i}(P)=0$| for all |$i$| if and only if the reading word of |$P$| is |$123\cdots n$|⁠. Thus, the coefficient of |$s_{(n)}$| in |$\text{rev}_{q}(\text{grFrob}(R_{n,k}))$| is

On the other hand, the coefficient of |$s_{(n)}$| in the charge formula of Theorem 4 is

To prove that (21) and (22) are equal via combinatorial methods, we first prove a lemma about charge, and then we define a bijection |$f$| from the set of tableaux |$T$| appearing in the sum (22) to the OSPs in (21) as follows.

An example of |$f(T)$| is depicted in Figure 2.

7 Skewing Formulas for the Delta Conjecture at Low t Degrees

It is natural to ask whether our skewing formula for the Delta Conjecture at |$t=0$| extends to the full Delta Conjecture symmetric function. In this section, we give several conjectures of such expansions below. Each formula may be expanded in order to obtain a positive Schur expansion.

Let |$[n]_{q} = 1+q + \cdots + q^{n-1}$| and |$\choose{n}{m}_{q} = \frac{[n]_{q}}{[m]_{q}[n-m]_{q}}$| be the usual |$q$|-analogues.

We have checked both Conjectures 15 and 16 computationally up to |$n=8$| for all |$k\leq n$|⁠. In the case of |$k=2$|⁠, we have the following formula for the full Delta Conjecture symmetric function.

Before we state the formula, we recall the Littlewood-Richardson rule for skew Schur functions in the case of two-row partitions. Given |$\lambda = (\lambda _{1},\lambda _{2})$| and |$\mu = (\mu _{1},\mu _{2})$| partitions,

where |$c_{\mu ,\nu }^{\lambda }$| is the number of semistandard Young tableaux |$T$| of skew shape |$\lambda /\mu $| with content |$\nu $| whose reverse reading word is Yamanouchi, meaning that if one reads the labels of |$T$| in reverse reading order, there are never more |$2$|s than |$1$|s up to any given point.

8 Next Directions

The new results and connections to geometry in this paper open up several natural directions for further investigation.

This paper rests in type A, but the Borho–MacPherson paper is type independent, so we also ask the following.

On the combinatorics side, since Corollaries 1.1 and 1.6 give formulas for the |$t=0$| specialization of the Delta Conjecture, and Section 7 gives conjectures for other |$t$| degrees, we also ask whether we can extend these formulas to the full Delta Conjecture symmetric functions for all |$t$| degrees.

Finally, the proofs in this paper rely heavily on the deep geometric, topological, and representation-theoretic machinery developed by Borho and MacPherson. We would like to see a combinatorial proof along the lines of the Lascoux–Schützenberger proof of the Hall-Littlewood cocharge formula (see [5] for a modern exposition of this proof).

The companion paper [11] will also investigate combinatorial routes towards Theorem 3 via a new formula in terms of Compositional Shuffle Theorem creation operators [6, 16].

Combining Theorem 1 and (17), our result gives a formula for the symmetric function |$s_{((n-k)^{s-1})}^{\perp }\widetilde{H}_{\Lambda }$| as a positive sum of Hall-Littlewood polynomials. Furthermore, by [9] there is also a formula for |$e_{j}^{\perp }\widetilde{H}_{\nu }$| for any |$j$| and |$\nu $| as a sum of Hall-Littlewood polynomials.

Acknowledgments

We thank Brendon Rhoades for inspiring conversations at the start of this work, and Jim Haglund for helpful feedback after a talk on this material. We also thank William Graham and Amber Russell for helpful conversations on partial resolutions.

References

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