1 Introduction
Delta and Theta operators, denoted by
$\Delta _F$
and
$\Theta _F$
for a choice of symmetric function F, are fundamental symmetric function operators in the theory of Macdonald polynomials. Since their introduction, these operators have been shown to have incredible properties and connections to other areas of interest. In introducing a brief history of these operators, we will point out some of these connections. For definitions of the symmetric functions discussed here, we refer the reader to Section 3. Often, this area of study has three aspects. There is the symmetric function side, the representation theoretical side, and a combinatorial description. By giving Schur function expansions of the symmetric function side, one is able to give the multiplicities of irreducible representations in the representation theoretical side via the Frobenius map, which sends irreducible characters of the symmetric group to Schur functions: Let
$A^{\lambda }$
denote Young’s irreducible representation of the symmetric group
$S_n$
indexed by the partition
$\lambda \vdash n$
. For any graded module
$$ \begin{align*} V = \bigoplus_{\alpha} V_{\alpha} & & \text{ with } & & V_\alpha \simeq \bigoplus_{\lambda \vdash n} n^{\alpha}_\lambda A^\lambda, \end{align*} $$
the graded Frobenius characteristic produces the symmetric function
$$\begin{align*}\mathcal{F}(V) = \sum_{\lambda \vdash n} s_\lambda \sum_{\alpha} n_\lambda^\alpha Q^\alpha. \end{align*}$$
On the other hand, when the combinatorial expansion of a symmetric function is Schur positive, it predicts the existence of a representation theoretical side. As we will describe here, when dealing with Macdonald polynomials, the representations associated to these expansions are often natural and important for a variety of areas of study.
As proved in [Reference HaimanHai02] and conjectured in [Reference Garsia and HaimanGH96],
$\Delta _{e_n} e_n$
gives the bigraded Frobenius characteristic for the space of
$S_n$
coinvariants of the polynomial ring with two sets of commuting variables. More precisely, if
$Y_n = y_1,\dots , y_n$
and
$Z_n = z_1,\dots , z_n$
are two sets of commuting variables, then
$\sigma \in S_n$
acts diagonally on the space of polynomials in
$Y_n,Z_n$
by sending
$y_i \mapsto y_{\sigma _i}$
and
$z_i \mapsto z_{\sigma _i}$
. The space of diagonal coinvariants is given by the quotient
$$\begin{align*}\mathcal{R}^{(2,0)} = \frac{\mathbb{C}[Y_n,Z_n]}{(\mathbb{C}[Y_n,Z_n]^{S_n}_+)}, \end{align*}$$
where
$(\mathbb {C}[Y_n,Z_n]^{S_n}_+)$
is the ideal generated by
$S_n$
-invariants with no constant term. This space is
$\mathbb {N}^2$
graded, and we can record the grading by setting
$Q^{(r,s)} = q^r t^s.$
Then Haiman’s theorem states that
$$\begin{align*}\Delta_{e_n} e_n = \mathcal{F} (\mathcal{R}^{(2,0)}). \end{align*}$$
The symmetric function
$\Delta _{e_n} e_n$
is most often denoted
$\nabla e_n$
, where
$\nabla $
is the Bergeron-Garsia nabla operator defined in [Reference Bergeron and GarsiaBG99]. Haiman proves this equality through algebraic geometrical means, realizing this ring through the Hilbert scheme of points on the plane. Hogancamp showed that the hook Schur functions in this symmetric function give the triply graded Khovanov-Rozansky homology for
$(n,n+1)$
-torus knots. There is a more general statement involving
$(n,nm \pm 1)$
torus knots, though we will not go into detail [Reference HogancampHog17].
On the combinatorial side, there is the Shuffle theorem, conjectured in [Reference Haglund, Haiman, Loehr, Remmel and UlyanovHHL+05b] and proved by Carlsson and Mellit [Reference Carlsson and MellitCM18]. This conjecture stated that
$\nabla e_n$
can be written as a sum over labeled Dyck paths. Carlsson and Mellit in fact prove the compositional refinement conjectured in [Reference Haglund, Morse and ZabrockiHMZ12] via the identity
$\nabla e_n = \sum _{\alpha \vDash n} \nabla C_\alpha $
. Their methods introduced a Dyck path algebra. Mellit expanded this idea in order to prove the related Rational Shuffle theorem [Reference MellitMel21], and then showed that the triply graded Khovanov-Rozansky homology for
$(m,n)$
-torus knots can be realized through the elliptic Hall or Schiffmann algebra [Reference MellitMel22]. On symmetric functions, this algebra can be generated by using the operators of multiplication by
$e_1$
and
$\Delta _{e_1}$
. Theta operators can also be viewed as elements of this algebra.
The Delta conjecture [Reference Haglund, Remmel and WilsonHRW18] gives a similar combinatorial description to the symmetric function
$\Delta ^{\prime}_{e_{n-k-1}} e_n$
. Soon after, Zabrocki [Reference ZabrockiZab19] gave a corresponding
$S_n$
-module for this symmetric function, stating that if we introduce a new set of anticommuting variables
$T_n = \tau _1,\dots , \tau _n$
, and set
$$\begin{align*}\mathcal{R}^{(2,1)} = \frac{\mathbb{C}[Y_n,Z_n, T_n]}{(\mathbb{C}[Y_n,Z_n,T_n]^{S_n}_+)}, \end{align*}$$
then
$\sum _{k} u^{k} \Delta _{e_{n-k-1}}' e_n$
gives
$\mathcal {F}(\mathcal {R}^{(2,1)} )$
, the triply graded Frobenius characteristic for the space of
$S_n$
coinvariants in two sets of commuting variables and one set of anticommuting variables. The methods used by Carlsson and Mellit in the proof of the shuffle theorem relied on the compositional refinement of the statement; however, the symmetric function
$\Delta ^{\prime}_{e_{n-k-1}} C_\alpha $
is not combinatorial. Theta operators were then introduced in [Reference D’Adderio, Iraci and WyngaerdDIVW21] in order to give a compositional refinement of the Delta conjecture, using that
$\Delta _{e_{n-k-1}}' e_n = \Theta _{e_k} \nabla e_{n-k}$
and the fact that
$\Theta _{e_k} \nabla C_\alpha $
is indeed combinatorial. This refinement ultimately led to a proof of the compositional Delta theorem [Reference D’Adderio and MellitDM22]. Most recently, the extended Delta conjecture was also proved in [Reference Blasiak, Haiman, Morse, Pun and SeelingerBHM+23], giving the combinatorial description for
$\Delta _{h_a} \Delta _{e_{k-1}}' e_n$
. This is realized through a connection to
$GL_m$
characters and the
$LLT$
polynomials of [Reference Lascoux, Leclerc and ThibonLLT97].
If we introduce yet another set of anticommuting variables and let
$\mathcal {R}^{(2,2)}$
be the
$S_n$
coinvariants with two sets of commuting and two sets of anticommuting variables, then it was also conjectured in [Reference D’Adderio, Iraci and WyngaerdDIVW21] that
$$ \begin{align} \mathcal{F}(\mathcal{R}^{(2,2)}) = \sum_{r,s \geq 0} u^r v^s \Theta_{e_r} \Theta_{e_ s} \nabla e_{n-r-s}, \end{align} $$
meaning the Frobenius characteristic of
$\mathcal {R}^{(2,2)}$
is given via Theta operators. The purely fermionic case
$\mathcal {R}^{(0,2)}$
, involving only the portion with anticommuting variables (obtained by setting
$q=t=0$
in (1.1)) has recently been proved in [Reference Iraci, Rhoades and RomeroIRR23]. For the
$\mathcal {R}^{(1,1)}$
case (found by setting
$t=u=0$
in (1.1)), the graded dimension of the coinvariant space with one set of commuting variables and one set of anticommuting variables has been shown in [Reference Rhoades and WilsonRW23] to agree with the conjectured formula.
Theta operators have shown remarkable positivity properties. In [Reference D’Adderio, Iraci, Le Borgne, Romero and WyngaerdDILB+22], the authors give a conjectural formula for
$\Theta _{e_\lambda } e_1$
when
$q=1$
, in terms of tiered trees, known as Theta Tree Conjecture. When
$\lambda $
has two parts, via [Reference D’Adderio, Iraci, Le Borgne, Romero and WyngaerdDILB+22, Theorem 7.2], this expression directly relates to the (conjectured) Frobenius characteristic of
$\mathcal {R}^{(2,2)}$
; when
$\lambda = 1^n$
, it was shown to be the generating function for the Kac polynomial of certain quivers, adding yet another geometrical meaning to symmetric functions arising in the study of Macdonald polynomials and its related operators. In the same work, the authors also give a very similar formula for
$M \Delta _{e_1} \Pi e_\lambda ^\ast $
, which is an expression that arises naturally when working with Theta operators; the analogous statement is known as the Symmetric Theta Tree Conjecture, and the combinatorial objects involved exhibit nicer symmetries.
This conjecture leads us to the study of the expression 
, and, in the same fashion as the Extended Delta Conjecture, to the more general expression
$\Delta _{m_\gamma } \operatorname {\mathrm {\Xi }} e_\lambda $
(and
$\Delta _{m_\gamma } \operatorname {\mathrm {\Xi }} s_\lambda $
). In this work, we show a positive e-expansion for these symmetric functions when
$t=1$
. One can hope that, exploiting the many symmetric function identities involving Theta operators [Reference D’Adderio and RomeroDR23], these results can directly relate to the aforementioned conjectures.
If a symmetric function is positive in some basis, then setting
$t=1$
(or
$q=1$
) leaves the ungraded multiplicities intact. Therefore, giving an expansion at
$t=1$
would predict the combinatorial objects enumerated by these symmetric functions without the specialization. Even more, we find that certain symmetric functions are not Schur positive, yet become positive in the elementary basis when
$t=1$
. And even more surprising, we have Conjecture 13.1, which predicts that the expression
$\Delta _{m_\gamma } \operatorname {\mathrm {\Xi }} s_\lambda $
is e-positive after substituting
$q=1+u$
(rather than substituting
$t=1$
).
The main strategy of our work is to expand the symmetric function, when
$t=1$
, as a series in q. One of the amazing aspects of this method, found in [Reference Hicks and RomeroHR18], is the use of the combinatorial formula for forgotten symmetric functions and their principal evaluation. The terms in the series are sums of certain signed combinatorial objects. After applying a weight-preserving, sign-reversing involution, we are able to get a finite number of positive fixed points, which bijectively correspond to some set of labeled polyominoes. The end result is found by adjusting the polyomino picture to get an expansion in terms of what we call
$\gamma $
-parking functions:
Theorem 1.1. For any two partitions
$\lambda $
and
$\gamma $
, there is a family of labeled polyominoes
${\mathrm { {PF}}}^{\gamma }_{\lambda }$
, called
$\gamma $
-parking functions of content
$\lambda $
, and a statistic
$\operatorname {\mathrm {area}}$
giving
$$ \begin{align} \left. \Delta_{m_\gamma} \operatorname{\mathrm{\Xi}} e_{\lambda} \right\rvert_{t=1} = \sum_{p \in {\mathrm{ {PF}}}^{\gamma}_{\lambda} } q^{\operatorname{\mathrm{area}}(p)} e_{\eta(p)}. \end{align} $$
This gives a combinatorial expansion for
$\operatorname {\mathrm {\Xi }} e_\lambda \rvert _{t=1}$
that is different from the one given in [Reference D’Adderio, Iraci, Le Borgne, Romero and WyngaerdDILB+22] in terms of tiered trees, and leaves the interesting problem of finding a correspondence between the two, which would be enough to bijectively prove the Symmetric Theta Conjecture.
Using the same methods that prove Theorem 1.1, we also show
Theorem 1.2. For any two partitions
$\lambda $
and
$\gamma $
, there is a family of labeled polyominoes
${\mathrm { {LPF}}}^{\gamma }_{\lambda }$
, called lattice
$\gamma $
-parking functions of content
$\lambda $
, and a statistic
$\operatorname {\mathrm {area}}$
giving
$$ \begin{align} \left. \Delta_{m_\gamma} \operatorname{\mathrm{\Xi}} s_{\lambda} \right\rvert_{t=1} = \sum_{p \in {\mathrm{ {LPF}}}^{\gamma}_{\lambda'} } q^{\operatorname{\mathrm{area}}(p)} e_{\eta(p)}. \end{align} $$
It is now natural to ask whether this expression also has an interpretation in terms of tiered trees, and if there are further generalizations of these identities.
2 Combinatorial definitions
In this section, we aim to introduce the combinatorial objects that will give us the symmetric function expansions we are interested in.
2.1 Words
Definition 2.1. A word of length r is an element
$w = (w_1, \dots , w_r) \in \mathbb {N}^r$
. We denote the length by 
and the size by 
.
Let w be a word of length r. We let
$m_i(w)$
be the number of indices j, such that
$w_j = i$
, that is,
$m_i(w)$
is the multiplicity of i in w; we denote the multiplicity type of w as
$m(w) = 0^{m_0(w)}1^{m_1(w)} 2^{m_2(w)} \cdots $
. If
$w \in \mathbb {N}^r_+$
(it has no
$0$
entries), then we call it a composition and write
$w \vDash \lvert w \rvert $
. There is a class of words that is of special interest to us.
Definition 2.2. A lattice word is a word
$w = (w_1, \dots , w_r) \in \mathbb {N}_+^r$
, such that, for all
$1 \leq i, j \leq r$
, we have
$$\begin{align*}m_{j+1}(w_1,\dots,w_i) \leq m_j(w_{1},\dots, w_{i}), \end{align*}$$
that is, a word, such that every prefix has at least as many
$1$
s as
$2$
s, at least as many
$2$
s as
$3$
s, and so on.
Denote by
$R(w)$
the set of all words
$\alpha = (\alpha _1, \dots , \alpha _r)$
whose entries can be rearranged to give w, or
$m(\alpha ) = m(w)$
. If
$\alpha _1 \geq \alpha _2 \geq \cdots \geq \alpha _r> 0$
, then
$\alpha $
is a partition, written
$\alpha \vdash \lvert \alpha \rvert $
. It will be convenient to write a sequence of words
$\vec {w}= (w^1,\dots , w^r)$
, with
$w^i \in \mathbb {N}^{r_i}$
, as a vector. The type of
$\vec {w}$
, denoted by
$m(\vec {w})$
, is the multiplicity type of the concatenation
$w^1 \cdots w^r = (w^1_1, w^1_2, \dots , w^2_1, w^2_2, \dots , \dots ).$
Definition 2.3. We define the sets of word vectors of length
$\beta $
and content
$\alpha $
, composition vectors of size
$\beta $
rearranging to
$\alpha $
, and partition vectors of size
$\beta $
rearranging to
$\alpha $
as
The first is the set of sequences of words where the collective multiplicity of i is
$\alpha _i$
and sequence j has length
$\beta _j$
. If
$\ell (\alpha ) < \lvert \beta \rvert $
, then it is impossible to do this without allowing
$0$
entries, of which there must be
$|\beta |-\ell (\alpha )$
. The second set is the sequence of compositions whose sizes are determined by
$\beta $
and whose parts collectively rearrange to
$\alpha $
; and the last set is the set of sequences of partitions whose sizes are determined by
$\beta $
and whose collective union of parts rearranges to
$\alpha $
.
Example 2.4. For
$\alpha = (1,1,2,1,3,1)$
,
$\beta = (3,1,5)$
, we have
$((1,2,5), (3), (4,3,5,5,6)) \in {\mathrm { {WV}}}(\alpha , \beta )$
,
$((2,1), (1), (1,3,1)) \in \operatorname {\mathrm {CR}}(\alpha , \beta )$
,
$((2,1), (1), (3,1,1)) \in \operatorname {\mathrm {PR}}(\alpha , \beta )$
.
Definition 2.5. We define the descent set of a word w as 
, and the ascent set as 
. We have the following statistics.
Note that
$\operatorname {\mathrm {revmaj}}$
and
$\operatorname {\mathrm {revcomaj}}$
are actually the
$\operatorname {\mathrm {maj}}$
and
$\operatorname {\mathrm {comaj}}$
of the reverse word, hence the name.
2.2
$\gamma $
-Dyck paths
We need to recall this classical definition.
Definition 2.6. A parallelogram polyomino of size
$m \times n$
is a pair of lattice paths
$(P,Q)$
from
$(0,0)$
to
$(m,n)$
, consisting of unit North and East steps, such that P (the top path) lies always strictly above Q (the bottom path), except on the endpoints.
Definition 2.7. The area of a parallelogram polyomino of size
$m \times n$
is defined as
Since the two paths P and Q do not touch between the endpoints,
$m+n-1$
is the minimal number of unit cells between them. An example is given in Figure 1.
Figure 1 A parallelogram polyomino with area
$20$
.
We can now introduce our new objects.
Definition 2.8. Let
$\gamma \vdash m$
. A
$\gamma $
-Dyck path of size n is a parallelogram polyomino of size
$(m + n + 1) \times n$
, such that the bottom path does not have two consecutive North steps, and if
$\alpha _i$
is the number of East steps of the bottom path on the line
$x = i-1$
, then
$(\alpha _1 - 1, \alpha _2, \dots , \alpha _n)$
rearranges to 
.
In other words, we start from a “staircase” path with two East steps on the x-axis and one East step on each other line
$x=i$
for
$1 \leq i < n$
, and we insert on each of these lines a number of East steps given by the parts of
$\gamma $
, in some order.
Remark 2.9. Notice that
$\varnothing $
-Dyck paths are essentially the same thing as classical Dyck paths. Indeed,
$\gamma = \varnothing $
implies
$m=0$
, so the polyomino is of size
$(n+1) \times n$
and the bottom path is the staircase as mentioned above. This given, the requirement that the two paths do not touch is exactly asking that the top path lies always weakly above the diagonal
$x=y$
(see Figure 2 for an example). The importance of this fact will be apparent in Section 12.2.
Figure 2 A
$\varnothing $
-Dyck path.
Figure 3 The set of
$(2)$
-parking functions of height
$2$
.
Definition 2.10. A labeled
$\gamma $
-Dyck path, is a
$\gamma $
-Dyck path in which each North step of the top path is assigned a positive integer label, such that consecutive North steps are assigned strictly increasing labels. A labeled
$\gamma $
-Dyck path will be denoted as a triple
$p = (P,Q,w)$
, where P is the top path, Q is the bottom path, and w is the word formed by the labels when read from bottom to top. The content of a labeled
$\gamma $
-Dyck path is the weak composition
$\alpha \vDash _w n$
whose parts
$\alpha _i$
give the number of i’s appearing in the labeling (or
$m(w) =0^{n-\ell (\alpha )}1^{\alpha _1}2^{\alpha _2}\dots $
). A
$\gamma $
-parking function is a labeled
$\gamma $
-Dyck path of content
$1^n$
. For our convenience, we will also refer to labeled
$\gamma $
-Dyck paths of content
$\alpha $
as
$\gamma $
-parking functions of content
$\alpha $
, and denote them by
${\mathrm { {PF}}}_\alpha ^\gamma $
(see Figure 2 for examples of
$\gamma $
-parking functions).
Definition 2.11. A lattice
$\gamma $
-Dyck path is a labeled
$\gamma $
-Dyck path in which the sequence of labels, read bottom to top, is a lattice word. Notice that the content of a lattice word is necessarily a partition. As above, for our convenience, we will also refer to lattice
$\gamma $
-Dyck paths with content
$\lambda $
as lattice
$\gamma $
-parking functions with content
$\lambda $
, and denote them by
${\mathrm { {LPF}}}_\lambda ^\gamma $
.
Definition 2.12. The e-composition
$\eta (p)$
of a labeled
$\gamma $
-Dyck path
$p = (P,Q,w)$
is defined as follows: Let
$\overline {P}$
be the path obtained from P by removing the first East step after the
$i^{\textit {h}}$
North step for every
$i \not \in \operatorname {\mathrm {Asc}}(w)$
;
$\eta (p)$
is the composition whose parts are the lengths of the maximal sequences of consecutive North steps appearing in
$\overline {P}$
, from the bottom to the top (see Figure 4 for an example).
Figure 4 The e-composition of a labeled
$\gamma $
-Dyck path. In this case,
$\eta (p) = (4,2,1)$
.
3 Symmetric function preliminaries
The standard reference for Macdonald polynomials is Macdonald’s book [Reference MacdonaldMac95]. For some reference on modified Macdonald polynomials, plethystic substitution, and Delta operators, we have [Reference HaglundHag08] and [Reference Bergeron, Garsia, Haiman and TeslerBGHT99]. As a reference for Theta operators, we have [Reference D’Adderio, Iraci and WyngaerdDIVW21] and [Reference D’Adderio and RomeroDR23].
We represent partitions by their Young diagram. For a partition
$\mu $
and a cell
$c \in \mu $
, we let
$a(c)$
,
$l(c)$
,
$a'(c)$
, and
$l'(c)$
denote the arm, leg, coarm, and coleg of the cell. This gives the number of cells in
$\mu $
strictly to the right, above, to the left, and below of c, respectively (see Figure 5 for an example).
Figure 5 Limbs and colimbs of a cell in a partition.
From here on, we set 
. For any partition
$\mu $
, we define the constants
Recall the ordinary Hall scalar product gives the orthogonality relation
where, for any proposition A,
$\chi (A) = 1$
if A is true, and
$0$
otherwise. The
$\ast $
-scalar product may be given by setting for any two symmetric functions F and G,
where
$\omega $
is the algebra isomorphism on symmetric functions defined by 
. Note that
$\omega $
is also an isometry and an involution.
The modified Macdonald basis is orthogonal with respect to the
$\ast $
-scalar product, that is,
$$ \begin{align} \langle \operatorname{\mathrm{\widetilde{H}}}_\lambda , \operatorname{\mathrm{\widetilde{H}}}_\mu \rangle_\ast = w_\mu \chi(\lambda = \mu). \end{align} $$
We recall the definition of Delta operators, which are eigenoperators of the modified Macdonald basis indexed by symmetric functions [Reference Bergeron, Garsia, Haiman and TeslerBGHT99].
Definition 3.1. For
$F \in \Lambda $
, we define the operator
$\Delta _{F} \colon \Lambda \rightarrow \Lambda $
by setting 
on the Macdonald basis and extending by linearity. We then define 
.
We now introduce the q-Pochhammer symbol.
Definition 3.2. For
$r \in \mathbb {N}$
, we define the q-Pochhammer symbol as
and for
$\mu $
a partition, we set
Remark 3.3. When
$t=1$
, the modified Macdonald basis specializes as
$$\begin{align*}\operatorname{\mathrm{\widetilde{H}}}_\mu[X;q,1] = (q;q)_\mu h_\mu \left[ \frac{X}{1-q} \right]; \end{align*}$$
this means that, on the space of symmetric functions with coefficients in
$\mathbb {Q}(q)$
(rather than
$\mathbb {Q}(q,t)$
), the operator
$\widetilde {\Delta }_F$
can be defined by setting
$$\begin{align*}\widetilde{\Delta}_F h_\mu\left[ \frac{X}{1-q} \right] = F\left[ \sum_{i=1}^{\ell(\mu)} [\mu_i]_q \right] h_\mu \left[ \frac{X}{1-q} \right]. \end{align*}$$
This specialization will be useful later.
We now define the Theta operators.
Definition 3.4 [Reference D’Adderio, Iraci and WyngaerdDIVW21, (28)].
For any homogeneous symmetric function
$F \in \Lambda $
, we define the Theta operators
$\Theta _F \colon \Lambda \rightarrow \Lambda $
as follows. For any homogeneous symmetric function
$G \in \Lambda $
, we set
We will often use the common shorthand
Also notice that, from the definition, one has
$\Theta _F + \Theta _G = \Theta _{F+G}$
and
$\Theta _{F} \Theta _G = \Theta _{FG}.$
In [Reference D’Adderio, Iraci, Le Borgne, Romero and WyngaerdDILB+22], the authors give a (conjectural) combinatorial formula for
$\Theta _{e_\lambda } e_1$
(when
$t=1$
) in terms of rooted tiered trees, and then a very similar formula for
$M \Delta _{e_1} \Pi e_\lambda ^{\ast }[X]$
(also when
$t=1$
) in terms of
$0$
-rooted tiered trees, which have nicer symmetries. The expression
$M \Delta _{e_1} \Pi e_\lambda ^{\ast }[X]$
seems to have surprising positivity properties and pops up in various symmetric function identities (cf. [Reference Bergeron, Haglund, Iraci and RomeroBHIR23]). For this reason, it is convenient to define the following.
Definition 3.5. We define the linear operator
$\operatorname {\mathrm {\Xi }} \colon \Lambda \rightarrow \Lambda $
as
In the remainder of this work, we will show another combinatorial expansion for
$\operatorname {\mathrm {\Xi }} e_\lambda \rvert _{t=1}$
, different from the one in [Reference D’Adderio, Iraci, Le Borgne, Romero and WyngaerdDILB+22]. We will actually prove a more general result, namely, an expansion for
$\widetilde {\Delta }_{m_\gamma } \operatorname {\mathrm {\Xi }} e_{\lambda }$
, which has the remarkable property of being e-positive. Without the specialization
$t=1$
, the expression
$\operatorname {\mathrm {\Xi }} e_\lambda $
is conjecturally Schur positive, but that fails in general when
$\gamma \neq \varnothing $
, making the global e-positivity when
$t=1$
even more remarkable.
4 Preliminary manipulations and specializations
In this section, we go through some algebraic manipulations, in order to give a combinatorial meaning to the symmetric function
$\widetilde {\Delta }_{m_\gamma } \operatorname {\mathrm {\Xi }} e_{\lambda }$
.
Lemma 4.1. For
$\lambda \vdash n$
and
$\gamma $
any partition, we have
$$\begin{align*}\Delta_{m_\gamma} \operatorname{\mathrm{\Xi}} e_\lambda = \sum_{\mu \vdash n} m_\gamma[B_\mu] \frac{ M B_\mu \Pi_\mu}{w_\mu} \langle h_{\lambda}, \operatorname{\mathrm{\widetilde{H}}}_\mu \rangle \operatorname{\mathrm{\widetilde{H}}}_\mu. \end{align*}$$
Proof. By (3.3) and (3.2), we have
$$\begin{align*}e_{\lambda}^\ast = \sum_{\mu \vdash n} \frac{\operatorname{\mathrm{\widetilde{H}}}_\mu}{w_\mu} \langle e_{\lambda}^\ast, \operatorname{\mathrm{\widetilde{H}}}_\mu \rangle_\ast = \sum_{\mu \vdash n} \frac{\operatorname{\mathrm{\widetilde{H}}}_\mu}{w_\mu} \langle h_{\lambda}, \operatorname{\mathrm{\widetilde{H}}}_\mu \rangle. \end{align*}$$
Now by Definition 3.5 and Definition 3.1, we have
$$\begin{align*}\Delta_{m_\gamma} \operatorname{\mathrm{\Xi}} e_\lambda = \Delta_{m_\gamma} M \Delta_{e_1} \Pi e_{\lambda}^\ast = \sum_{\mu \vdash n} m_\gamma[B_\mu] \frac{ M B_\mu \Pi_\mu}{w_\mu} \langle h_{\lambda}, \operatorname{\mathrm{\widetilde{H}}}_\mu \rangle\operatorname{\mathrm{\widetilde{H}}}_\mu, \end{align*}$$
as desired.
We will break up these summation terms by analyzing each of the three factors, specializing t to
$1$
for each one individually.
Lemma 4.2. For
$\mu \vdash n$
, we have
$$\begin{align*}\left. \frac{M B_\mu \Pi_\mu}{w_\mu} \right\rvert_{t=1} = (-1)^{n-\ell(\mu)} (q;q)_\mu^{-1} \sum_{\alpha \in R(\mu)} (1-q^{\alpha_1}). \end{align*}$$
Proof. It follows immediately from the definition that
$B_\mu \rvert _{t=1} = \sum _{i=1}^{\ell (\mu )} [\mu _i]_q$
. Let
$$\begin{align*}A_1 = \frac{ (1-q) \prod_{ \substack{ c \in \mu \\ a'(c) \neq 0}} (1-q^{a'(c)}t^{l'(c)})} {\prod_{\substack {c \in \mu \\ a(c) \neq 0}} (q^{a(c)} -t^{l(c)+1}) \prod_{c \in \mu} (t^{l(c)} -q^{a(c)+1}) } \end{align*}$$
and
$$\begin{align*}A_0 = \frac{(1-t) \prod_{ \substack{ c \in \mu \\ a'(c) = 0}} (1-q^{a'(c)}t^{l'(c)})}{\prod_{\substack{c\in \mu \\ a(c) = 0}} (q^{a(c)} -t^{l(c)+1}) }. \end{align*}$$
We have that
$M \Pi _\mu / w_\mu = A_1 \cdot A_0$
, where
$A_1$
collects the terms that do not vanish when
$t=1$
, and
$A_0$
collects the terms that evaluate to
$0$
when
$t=1$
.
Now, evaluating
$A_1$
, we get
$$\begin{align*}A_1 \rvert_{t=1} = \frac{(1-q) \prod_{\substack{c \in \mu \\ a'(c) \neq 0}} (1-q^{a'(c)})} {\prod_{\substack {c \in \mu \\ a(c) \neq 0}} (q^{a(c)} -1) \prod \limits_{c \in \mu} (1 -q^{a(c)+1})}, \end{align*}$$
and since
$$ \begin{align*} \frac{ \prod_{\substack{c \in \mu \\ a'(c) \neq 0}} (1-q^{a'(c)}) } { \prod_{\substack {c \in \mu \\ a(c) \neq 0}} (q^{a(c)} -1) } = (-1)^{n-\ell(\mu)} && \text{ and } && \prod_{c \in \mu} (1 -q^{a(c)+1}) = (q;q)_\mu, \end{align*} $$
we have
$$\begin{align*}A_1 \rvert_{t=1} = (-1)^{n-\ell(\mu)} \frac{(1-q)}{(q;q)_\mu}. \end{align*}$$
To evaluate
$A_0$
, notice first that
$$\begin{align*}A_0 = \frac{(1-t) \prod_{\substack{c \in \mu \\ a'(c) = 0}} (1-t^{l'(c)}) }{\prod_{\substack{c \in \mu \\ a(c) = 0}} (1 -t^{l(c)+1}) } = \frac{(1-t) (t;t)_{\ell(\mu)-1}}{(t;t)_{m_1(\mu)} \cdots (t;t)_{m_n(\mu)}} = \frac{1}{[\ell(\mu)]_t} {\genfrac{[}{]}{0pt}{}{{\ell(\mu)}}{{m_1(\mu), \dots, m_n(\mu)}}}_t, \end{align*}$$
where
$m_i(\mu )$
is the multiplicity of i in
$\mu $
. When we set
$t=1$
, we get the usual cyclic multinomial.
Putting the pieces together, we get
$$\begin{align*}\left. \frac{M B_\mu \Pi_\mu}{w_\mu} \right\rvert_{t=1} = \frac{(-1)^{n-\ell(\mu)}}{\ell(\mu)} \binom{\ell(\mu)}{ m_1(\mu), \dots, m_n(\mu)} \frac{(1-q)}{(q;q)_\mu} \sum_{i=1}^{\ell(\mu)} [\mu_i]_q. \end{align*}$$
Now we can interpret this product combinatorially. First, note that
$$\begin{align*}(1-q) \sum_{i=1}^{\ell(\mu)} [\mu_i]_q = \sum_{i=1}^{\ell(\mu)} (1-q ^{\mu_i}), \end{align*}$$
and that
$$\begin{align*}\binom{\ell(\mu) }{ m_1(\mu),\dots, m_n(\mu)} = \# R(\mu) \end{align*}$$
is the number of rearrangements
$\alpha =(\alpha _1,\dots , \alpha _{\ell }) \in \operatorname {\mathrm {R}}(\mu )$
of the parts of
$\mu $
. Therefore,
$$\begin{align*}\# R(\mu) \cdot \sum_{i=1}^{\ell(\mu)} (1-q^{\mu_i}) = \sum_{\alpha \in \operatorname{\mathrm{R}}(\mu)} \sum_{i=1}^{\ell(\mu)}( 1-q^{\alpha_i}). \end{align*}$$
This corresponds to selecting a rearrangement
$\alpha $
of
$\mu $
then selecting some i from
$1$
to
$\ell (\mu )$
. Equivalently, we can first select a rearrangement r, take
$1-q^{\alpha _1}$
, then circularly rearrange
$\alpha $
, keeping this selection of
$1-q^{\alpha _1}$
. Since there are
$\ell (\mu )$
circular rearrangements, we have that
$$\begin{align*}\frac{\# R(\mu)}{\ell(\mu)} \sum_{i=1}^{\ell(\mu)} (1-q^{\mu_i}) = \sum_{\alpha \in R(\mu)} (1-q^{\alpha_1}) \end{align*}$$
and so we can conclude that
$$\begin{align*}\left. \frac{MB_\mu \Pi_\mu}{w_\mu} \right\rvert_{t=1} = (-1)^{n-\ell(\mu)}(q;q)_{\mu}^{-1}\sum_{\alpha \in R(\mu)} (1-q^{\alpha_1}), \end{align*}$$
as desired.
Lemma 4.3. For
$\lambda ,\mu \vdash n$
, we have
$$\begin{align*}\left. \langle \operatorname{\mathrm{\widetilde{H}}}_\mu, h_\lambda \rangle \right\rvert_{t=1} = \sum_{\vec{w} \in {\mathrm{ {WV}}}(\lambda, \mu)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})}. \end{align*}$$
Proof. We recall the classical result (see [Reference StanleySta99] and [Reference HaglundHag08]) that
$$ \begin{align*} (q;q)_n h_n\left[\frac{X}{1-q} \right] & = \sum_{w= (w_1,\dots,w_n) \in \mathbb{N}_+^n} q^{\operatorname{\mathrm{maj}}(w)} x_{w_1} \cdots x_{w_n} \\ & = \sum_{w= (w_1,\dots,w_n) \in \mathbb{N}_+^n} q^{\operatorname{\mathrm{comaj}}(w)} x_{w_1} \cdots x_{w_n} \\ & = \sum_{w= (w_1,\dots,w_n) \in \mathbb{N}_+^n} q^{\operatorname{\mathrm{revmaj}}(w)} x_{w_1} \cdots x_{w_n}. \end{align*} $$
For our purposes, we need the last of these equalities, involving the reverse major index. Recall that
$\langle h_\lambda , m_\mu \rangle = \chi (\lambda =\mu )$
. Since the homogeneous basis is multiplicative, this means that
$$\begin{align*}\left\langle {(q;q)}_\mu h_\mu \left[ \frac{X}{1-q} \right] , h_\lambda \right\rangle = \left\langle \prod_{i=1}^{\ell(\mu)} \sum_{w \in \mathbb{N}_+^{\mu_i}} q^{\operatorname{\mathrm{revmaj}}(w)} x_{w_1} \cdots x_{w_{\mu_i}}, h_\lambda \right\rangle = \sum_{ \vec{w} \in {\mathrm{ {WV}}}(\lambda, \mu)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})}, \end{align*}$$
where
$\operatorname {\mathrm {revmaj}}(\vec {w})= \operatorname {\mathrm {revmaj}}(w^1)+\cdots + \operatorname {\mathrm {revmaj}}(w^n)$
. Now by Remark 3.3, we have
$$\begin{align*}\left. \langle \operatorname{\mathrm{\widetilde{H}}}_\mu, h_\lambda \rangle \right\rvert_{t=1} = \sum_{\vec{w} \in {\mathrm{ {WV}}}(\lambda, \mu)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})}, \end{align*}$$
as desired.
Let us recall the Cauchy identity.
Proposition 4.4 (Cauchy identity).
For any two expressions
$X,Y$
, and any two dual bases
$\{u_\lambda \}_\lambda $
,
$\{v_\lambda \}$
under the Hall scalar product, we have
$$\begin{align*}h_{n}[XY] = \sum_{\lambda \vdash n} u_{\lambda}[X] v_\lambda[Y]. \end{align*}$$
We now get to the final term of our product.
Lemma 4.5. For
$\mu \vdash n$
, we have
$$\begin{align*}\operatorname{\mathrm{\widetilde{H}}}_\mu[X;q,1] = {(q;q)_\mu} \sum_{\eta \vdash n} e_\eta[X] \sum_{ \vec{\nu} \in \operatorname{\mathrm{PR}}(\eta,\mu)} f_{\vec{\nu}}\left[ \frac{1}{1-q} \right]. \end{align*}$$
Proof. Recall that (see Remark 3.3)
$$\begin{align*}\operatorname{\mathrm{\widetilde{H}}}_\mu[X;q,1] = {(q;q)_\mu} h_\mu \left[ \frac{X}{1-q} \right]. \end{align*}$$
Now by Proposition 4.4, using the fact that the elementary symmetric functions and forgotten symmetric functions are dual, we have
$$ \begin{align} h_{n} \left[ \frac{X}{1-q} \right] = \sum_{\lambda \vdash n} e_\lambda[X] f_\lambda \left[ \frac{1}{1-q} \right]. \end{align} $$
Applying (4.1) to each factor in
$h_\mu [X/(1-q)]$
and collecting
$e_\eta $
terms, we obtain
$$\begin{align*}\operatorname{\mathrm{\widetilde{H}}}_\mu[X;q,1] = {(q;q)_\mu} \sum_{\eta \vdash n} e_\eta[X] \sum_{ \vec{\nu} \in \operatorname{\mathrm{PR}}(\eta,\mu)} f_{\vec{\nu}}\left[ \frac{1}{1-q} \right], \end{align*}$$
where
$f_{\vec {\nu }} = f_{\nu ^1} f_{\nu ^2} \cdots f_{\nu ^{\ell (\mu )}}$
.
Putting everything together, we get the following.
Proposition 4.6. For any
$\lambda \vdash n$
and
$\gamma $
any partition, we have
$$\begin{align*}\widetilde{\Delta}_{m_\gamma} \operatorname{\mathrm{\Xi}} e_{\lambda} = \sum_{\eta \vdash n} D_{\lambda, \eta}^{\gamma}(q) e_\eta, \end{align*}$$
where
$$\begin{align*}D^{\gamma}_{\lambda, \eta} = \sum_{\beta \vDash n} \sum_{\vec{w} \in {\mathrm{ {WV}}}(\lambda, \beta)} \sum_{\vec{\nu} \in \operatorname{\mathrm{PR}}(\eta,\beta)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} m_\gamma\left[ \sum_i [\beta_i]_q \right] (-1)^{n-\ell(\beta)} (1-q^{\beta_1}) f_{\vec{\nu}}\left[ \frac{1}{1-q} \right]. \end{align*}$$
Proof. By combining Lemma 4.1, 4.2, 4.3, and 4.5, we get the expansion
$$ \begin{align*} {\widetilde{\Delta}}_{m_{\gamma}} \operatorname{\mathrm{\Xi}} e_{\lambda} & = \sum_{\eta \vdash n} e_{\eta} \sum_{\mu \vdash n} (-1)^{n-\ell(\mu)} \sum_{\beta \in \operatorname{\mathrm{R}} (\mu)} (1-q^{\beta_1}) m_{\gamma} \left[\sum_i [\beta_i]_q \right] \\ & \quad \times \sum_{\vec{w} \in {\mathrm{ {WV}}} (\lambda, \mu)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} \sum_{\vec{\nu} \in \operatorname{\mathrm{PR}}(\eta,\mu)} f_{\vec{\nu}} \left[\frac{1}{1-q}\right]. \end{align*} $$
Now, instead of summing over all
$\mu $
then summing over all compositions
$\beta \in R(\mu )$
that rearrange to
$\mu $
, we can instead just sum over all compositions
$\beta $
, and the thesis follows.
Corollary 4.7. For any
$\lambda \vdash n$
, we have
$$\begin{align*}\left. \operatorname{\mathrm{\Xi}} e_{\lambda} \right\rvert_{t=1} = \sum_{\eta \vdash n} \sum_{\beta \vDash n} \sum_{\vec{w} \in {\mathrm{ {WV}}} (\lambda, \beta)} \sum_{\vec{\nu} \in \operatorname{\mathrm{PR}}(\eta,\beta)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} (-1)^{n-\ell(\beta)} (1-q^{\beta_1}) f_{\vec{\nu}} \left[ \frac{1}{1-q} \right] e_{\eta}. \end{align*}$$
Proof. Just set
$\gamma = \varnothing $
in Proposition 4.6.
5 Forgotten symmetric functions
For
$\mu \vdash n$
of length
$\ell $
, the combinatorial formula for the forgotten symmetric function
$f_\mu $
[Reference Egecioglu and RemmelER91] is given by
$$\begin{align*}f_\mu\left[ X \right] = (-1)^{n-\ell} \sum_{ \alpha \in \operatorname{\mathrm{R}}(\mu)} \sum_{i_1 \leq \cdots \leq i_{\ell}} x_{i_1}^{\alpha_1} \cdots x_{i_\ell}^{a_\ell}. \end{align*}$$
Now, substituting
$X = (1-q)^{-1}$
, we get the expansion
$$ \begin{align} f_\mu\left[ \frac{1}{1-q}\right] = (-1)^{n-\ell} \sum_{ \alpha \in \operatorname{\mathrm{R}}(\mu)} \sum_{0 \leq i_1 \leq \cdots \leq i_{\ell}} \left( q^{i_1} \right)^{\alpha_1} \cdots \left( q^{i_\ell} \right)^{a_\ell}. \end{align} $$
Definition 5.1. Let
$\mu \vdash n$
. A column-composition tableau of type
$\mu $
is a pair
$C = (\alpha , c)$
, where
$\alpha \in \operatorname {\mathrm {R}}(\mu )$
is a composition that rearranges to
$\mu $
, and
$c = (c_1 \leq c_2 \leq \dots \leq c_n)$
is a sequence, such that
$$\begin{align*}c_i < c_{i+1} \implies i \in \{\alpha_1,\alpha_1+\alpha_2, \dots, \alpha_1+\cdots + \alpha_{\ell-1}\}. \end{align*}$$
We denote by
$\operatorname {\mathrm {CC}}_\mu $
the set of column-composition tableaux of type
$\mu $
, and by
$\overline {\operatorname {\mathrm {CC}}}_\mu $
the subset of those, such that
$c_1 = 0$
. For
$C \in \operatorname {\mathrm {CC}}_\mu $
, we define the length of C as
$\ell (C) = \lvert \mu \rvert $
and size of C as
$\lvert C \rvert = c_1 + c_2 + \dots + c_n$
. We will write
$c_i(C)$
for
$c_i$
when we need to specify the column-composition tableau.
We can depict the elements of
$\operatorname {\mathrm {CC}}_\mu $
as follows.
-
1. First, draw a row of size
$\lvert \mu \rvert $
which we call the base, and then depict the composition
$\alpha \in \operatorname {\mathrm {R}}(\mu )$
by separating the columns of the base with vertical bars; for instance, when
$\mu = (3,2,2,2,1,1,1)$
and
$\alpha = (2,1,2,1,2,1,3)$
, we draw the base as -
2. Next, draw
$c_i$
cells above the
$i^{\textit {h}}$
column of the base; in continuing our example, if
$c = (0,0,0,1,1,1,1,1,3,3,3,3)$
, we draw it as
Let us define the q-enumerators
which are power series in q. Then, by construction, we have the following.
Proposition 5.2. For any partition
$\mu $
, we have
$$ \begin{align*} f_{\mu}\left[ \frac{1}{1-q} \right] = (-1)^{|\mu| - \ell(\mu)} {\mathbf{CC}}_\mu & & \text{ and } & & (1-q)^{\lvert \mu \rvert} f_{\mu}\left[ \frac{1}{1-q} \right] = (-1)^{|\mu| - \ell(\mu)} \overline{{\mathbf{CC}}}_\mu. \end{align*} $$
Proof. In Equation (5.1), each term in the principal evaluation of
$f_\mu $
is given by selecting a rearrangement
$\alpha $
of
$\mu $
, and choosing
$i_1\leq \cdots \leq i_{\ell (\mu )}$
. This uniquely determines an element
$(\alpha ,c) \in \operatorname {\mathrm {CC}}_\mu $
, where the first
$\alpha _1$
columns
$c_1,\dots , c_{\alpha _1}$
are of size
$i_1$
, the next
$\alpha _2$
columns
$c_{\alpha _1+1},\dots , c_{\alpha _1 + \alpha _2}$
are of size
$i_2$
, and so on. Since then
$$\begin{align*}\sum_{i=1}^{|\mu|} c_i = \sum_{j=1}^{\ell(\mu)} \alpha_j i_j, \end{align*}$$
we see that
$q^{\lvert (\alpha ,c) \rvert }$
equals the term in Equation (5.1) corresponding to choosing
$\alpha $
and
$i_1\leq \cdots \leq i_{\ell (\mu )}$
.
The second equality follows from the fact that if
$(\alpha ,c) \in CC_\mu $
, then so is
$(\alpha ,c+1^n)$
, where
$c+1^n = (c_1+1,\dots ,c_n+1)$
. This defines an injective map, and we have
$$\begin{align*}q^{ \lvert \mu \rvert} {\mathbf{CC}}_\mu = \sum_{ \substack{ C \in \operatorname{\mathrm{CC}}_\mu \\ c_1(C)>0 }} q^{\lvert C \rvert}. \end{align*}$$
Therefore
$$\begin{align*}{\mathbf{CC}}_\mu -q^{\lvert \mu \rvert} {\mathbf{CC}}_\mu = \sum_{ \substack{ C \in \operatorname{\mathrm{CC}}_\mu \\ c_1 = 0} }q^{\lvert C \rvert}, \end{align*}$$
which gives the last equality in the proposition.
We conclude this section with the following results.
Lemma 5.3. For any
$\vec {\nu } \in \operatorname {\mathrm {PR}}(\eta ,\beta )$
, we have
$$\begin{align*}(-1)^{n-\ell(\beta)} (1-q^{\beta_1}) f_{\vec{\nu}} \left[ \frac{1}{1-q} \right] = (-1)^{\ell(\eta) - \ell(\beta)} \overline{{\mathbf{CC}}}_{\nu^1} {{\mathbf{CC}}}_{\nu^2} \cdots {{\mathbf{CC}}}_{\nu^{\ell(\beta)}}. \end{align*}$$
Proof. Using Proposition 5.2, we can replace each
$f_\nu [1/(1-q)]$
with
${\mathbf {CC}}_\nu $
. Since
$\nu _1 \vdash \beta _1$
, we can also replace
$(1-q^{\beta _1}) {\mathbf {CC}}_{\nu _1}$
with
$\overline {{\mathbf {CC}}}_{\nu ^1}$
. Finally, since
$\ell (\nu ^1) + \cdots + \ell (\nu ^{\ell (\beta )}) = \ell (\eta )$
, we have
$$\begin{align*}(-1)^{n - \ell(\beta)} (-1)^{\ell(\nu^1) - \lvert \nu^1 \rvert} \cdots (-1)^{\ell(\nu^{\ell(\beta)}) - \lvert \nu^{\ell(\beta)} \rvert} = (-1)^{\ell(\eta)- \ell(\beta)}, \end{align*}$$
and the thesis follows.
Proposition 5.4. For
$\eta , \lambda \vdash n$
and
$\gamma $
any partition, we have
$$\begin{align*}D^{\gamma}_{\lambda, \eta} = \sum_{\beta \vDash n} \sum_{\vec{w} \in {\mathrm{ {WV}}}(\lambda, \beta)} \sum_{\vec{\nu} \in \operatorname{\mathrm{PR}}(\eta,\beta)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} m_\gamma\left[ \sum_i [\beta_i]_q \right] (-1)^{\ell(\eta) - \ell(\beta)} \overline{{\mathbf{CC}}}_{\nu^1} {{\mathbf{CC}}}_{\nu^2} \cdots {{\mathbf{CC}}}_{\nu^{\ell(\beta)}}. \end{align*}$$
6 Combinatorial expansions
Recall that we are trying to compute the coefficient in the expansion
$\widetilde {\Delta }_{m_\gamma } \operatorname {\mathrm {\Xi }} e_\lambda = \sum _{\eta } D^\gamma _{\lambda ,\eta } e_\eta $
, using the formula in Proposition 5.4. We interpret the terms showing up there by labeling a sequence of column-composition tableaux.
Definition 6.1. A labeled column-composition tableaux is a triple
$(C, w, l)$
, where C is a column-composition tableau,
$w \in \mathbb {N}_+^{\ell (C)}$
, and
$l \in \mathbb {N}^{\ell (C)}$
.
Definition 6.2. Let
$\lambda , \eta \vdash n$
, and
$\gamma \vdash m$
, such that
$\ell (\gamma ) \leq n$
. A sequence of labeled column-composition tableaux of type
$\lambda , \eta , \gamma $
is a tuple of labeled column-composition tableaux
$(C^i, w^i, l^i)_{1 \leq i \leq r}$
, such that, for
$\beta = (\beta _1, \dots , \beta _r)$
,
$\beta _i = \ell (C^i)$
, we have:
-
1.
$C^1 \in \overline {\operatorname {\mathrm {CC}}_{\nu ^1}}$
and
$C^i \in \operatorname {\mathrm {CC}}_{\nu ^i}$
for
$i>1$
, for some
$\vec {\nu } \in \operatorname {\mathrm {PR}}(\eta , \beta )$
; -
2.
$\vec {w} = (w^1, \dots , w^r) \in {\mathrm { {WV}}}(\lambda , \beta )$
; -
3.
$\vec {l} = (l^1, \dots , l^r) \in {\mathrm { {WV}}}(m(\gamma ), \beta )$
.
In other words, a sequence of labeled column-composition tableaux of type
$\lambda , \eta , \gamma $
, is a tuple of column-composition tableaux of sizes
$\beta _1, \dots , \beta _r$
, such that
$c_1(C^1) = 0$
, so that to each tableau we associate a partition
$\nu ^i \vdash \beta _i$
and two words
$w^i, l^i$
, such that
$\vec {\nu }$
rearranges to
$\eta $
, the global content of
$\vec {w}$
is given by
$\lambda $
, and the letters of l are the parts of
$\gamma $
followed by an appropriate number of trailing zeros.
We denote by
${\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
the set of sequences of column-composition tableaux of type
$\lambda , \eta , \gamma $
. For
$T = (T_i)_{1 \leq i \leq r} \in {\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
, we set
$w(T_i) = w^i$
and
$l(T_i) = l^i$
.
Definition 6.3. For
$T = (T_i)_{1 \leq i \leq r} \in {\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
, with
$T_i = (C^i, w^i, l^i)$
, let
$\nu ^i$
be the type of
$C^i$
, let 
, and let
We define
Notice that, for every letter in
$w^i$
or
$l^i$
, its contribution to the weight only depends on the letter itself and the number of letters to its right. Also notice that the sign is given by the parity of the number of vertical bars in
$C^i$
. Finally, we define
Example 6.4. We are now going through an example in full detail. Let
$\lambda =(3,2,2,2)$
,
$\eta =(3,2,1,1,1,1)$
,
$\gamma =(4,3,2,2,1)$
, so
$\lvert \lambda \rvert = \lvert \eta \rvert = 9$
and
$\ell (\gamma ) = 5 \leq 9$
. For our convenience, we add four trailing zeros to
$\gamma $
, so
$\gamma = (4,3,2,2,1,0,0,0,0)$
.
To build an element of
${\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
, first choose
$\beta \vDash 9$
, such that some permutation of
$\eta $
refines
$\beta $
, say
$\beta = (3,1,5)$
. Next, select
$\nu ^i \vdash \beta _i$
, say
$\nu ^1 = (2,1)$
,
$\nu ^2 = (1)$
,
$\nu ^3 = (3,1,1)$
, so that the union of parts is
$\eta $
. Then, pick
$C^1 \in \overline {\operatorname {\mathrm {CC}}}_{\nu ^1}$
and
$C^i \in \operatorname {\mathrm {CC}}_{\nu ^i}$
for
$i> 1$
; say, for example
(note that
$c_1(C^1)=0$
). Since
$\lambda = (3,2,2,2)$
, we have
$m(\lambda ) = (1,1,1,2,2,3,3,4,4)$
. Pick any permutation of it and split it into parts of lengths given by the sizes of the parts of
$\beta $
, say, for example
$\vec {w} = ((2,1,2),(4),(3,4,1,3,1))$
. Write these words into the bases of the tableaux. We get
Finally, pick any permutation of the parts of
$\gamma $
, and, again, split it into parts of lengths given by the sizes of the parts of
$\beta $
, say
$\vec {l} = ((0,2,0),(1),(2,0,4,3,0))$
. Write it underneath the bases of the tableaux. We get
which is an element of
${\mathrm { {LC}}}_{(3,2,2,2),(3,2,1,1,1,1)}^{(4,3,2,2,1)}$
We now want to compute the weight and the sign of this sequence. Since there are three vertical bars, we have that the sign is given by
$(-1)^3$
.
We can compute the weight of this sequence of labeled column-composition tableaux in three steps. First count the number of cells above the base rows: there are
$2$
,
$1$
, and
$5+4=9$
cells, respectively, so the total weight given by the cells is
$12$
.
The weight corresponding to the labels in the base is found by taking the reverse major index of each individual base. We compute this by taking
$$\begin{align*}\sum_{ w^i_j < w^i_{j+1}} \# \{ \text{cells in the same base and on the right of}\ w^i_j \}. \end{align*}$$
In the above example, we have that
$w^1 = (2,1,2)$
has an ascent in position
$2$
, and there is one cell to its right. Therefore,
$\operatorname {\mathrm {revmaj}}(w^1) = 1$
. Since
$w^2$
has length
$1$
, it has no ascents. Finally,
$w^3 = (3,4,1,3,1)$
has an ascent in position
$1$
and one in position
$3$
. There are four cells to the right of the label in position
$2$
, and two cells to the right of the last ascent. Therefore,
$\operatorname {\mathrm {revmaj}}(w^3) = 4+2$
, and the total contribution given by
$\vec {w}$
is
$7$
.
The last step is to calculate the contribution of labels underneath the base rows. For this, we will say a label
$l^i_j$
has
$\beta _i-j$
cells on its right, since these are the number of cells in its base row to the right of the label. We take
$$\begin{align*}\sum_{i,j} l^i_j \times \# \{ \text{cells in the same base and on the right of}\ l^i_j \}. \end{align*}$$
The first nonzero label from the left is a
$2$
in the second column of
$T_1$
. There is one cell to its right, meaning this label contributes by
$2 \cdot 1$
to the weight. The next nonzero label is a
$1$
, but there are no cells to its right, so its contribution is
$1 \cdot 0 = 0$
. Similarly,
$T_2$
has size
$1$
, so its contribution is also
$0$
. Finally, in
$T_3$
, the first label
$2$
has four cells to its right (so it contributes
$2 \cdot 4$
). The next label is a
$4$
and it has two cells to its right (so it contributes
$4 \cdot 2$
). The last nonzero label is a
$3$
and it has one cell to its right (so it contributes
$3 \cdot 1$
). Therefore, the labels under the base rows collectively contribute a factor of
$q^{2+8+8+3}$
to the weight. Putting everything together, we have
$$ \begin{align*} \operatorname{\mathrm{weight}}(T) = (2+1+9) + (1+0+4+2) + (2+8+8+3) = 40 \end{align*} $$
and
$\operatorname {\mathrm {sign}}(T) = (-1)^3 = -1$
.
The following proposition is an immediate consequence of our construction.
Proposition 6.5. For
$\eta , \lambda \vdash n$
and
$\gamma $
any partition, we have
$$\begin{align*}D^\gamma_{\lambda, \eta} = \sum_{T \in {\mathrm{ {LC}}}_{\lambda, \eta}^\gamma} q^{\operatorname{\mathrm{weight}}(T)} \operatorname{\mathrm{sign}}(T). \end{align*}$$
Proof. Recall that, by Proposition 5.4, we have
$$\begin{align*}D^{\gamma}_{\lambda, \eta} = \sum_{\beta \vDash n} \sum_{\vec{w} \in {\mathrm{ {WV}}}(\lambda, \beta)} \sum_{\vec{\nu} \in \operatorname{\mathrm{PR}}(\eta,\beta)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} m_\gamma\left[ \sum_i [\beta_i]_q \right] (-1)^{\ell(\eta) - \ell(\beta)} \overline{{\mathbf{CC}}}_{\nu^1} {{\mathbf{CC}}}_{\nu^2} \cdots {{\mathbf{CC}}}_{\nu^{\ell(\beta)}}. \end{align*}$$
Fix now a composition
$\beta \vDash n$
, such that some permutation of
$\eta $
refines
$\beta $
, a word vector
$\vec {w} \in {\mathrm { {WV}}}(\lambda , \beta )$
, and a partition vector
$\vec {\nu } \in \operatorname {\mathrm {PR}}(\eta ,\beta )$
. We want to study the summand
$$ \begin{align} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} m_\gamma\left[ \sum_i [\beta_i]_q \right] (-1)^{\ell(\eta)-\ell(\beta)} \overline{{\mathbf{CC}}}_{\nu^1} {\mathbf{CC}}_{\nu^2} \cdots {\mathbf{CC}}_{\nu^{\ell(\beta)}}. \end{align} $$
From the combinatorial formula for the monomial symmetric functions, we have
$$ \begin{align} m_\gamma\left[ [\beta_1]_q + \cdots + [\beta_{\ell(\beta)}]_q \right] = \sum_{\vec{l} \in {\mathrm{ {WV}}}(m(\gamma),\beta)} q^{u(l^1)+\cdots + u (l^{\ell(\beta)})}. \end{align} $$
Substituting (6.2) in (6.1), we get
$$\begin{align*}\sum_{\vec{l} \in {\mathrm{ {WV}}}(m(\gamma),\beta)} (-1)^{\ell(\eta)-\ell(\beta)} q^{u(l^1)+\cdots + u (l^{\ell(\beta)})} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} \overline{{\mathbf{CC}}}_{\nu^1} {\mathbf{CC}}_{\nu^2} \cdots {\mathbf{CC}}_{\nu^{\ell(\beta)}}, \end{align*}$$
and now every monomial in this expansion corresponds to a choice of
$\vec {l} \in {\mathrm { {WV}}}(m(\gamma ),\beta )$
,
$C^1 \in \overline {\operatorname {\mathrm {CC}}}_{\nu ^1}$
, and
$C^i \in \operatorname {\mathrm {CC}}_{\nu ^i}$
for
$i>1$
, giving the term
$$ \begin{align*} (-1)^{\ell(\eta)-\ell(\beta)} q^{\operatorname{\mathrm{revmaj}}(\vec{w})} & q^{u(l^1)+\cdots + u (l^{\ell(\beta)})} q^{\lvert C^1 \rvert + \cdots + \lvert C^{\ell(\beta)} \rvert} = \prod_{i=1}^{\ell(\beta)} (-1)^{\ell(\nu^i)-1} q^{\operatorname{\mathrm{revmaj}}(w^i) + u(l^i) + \lvert C^i \rvert}. \end{align*} $$
For each summand, let now
$T \in {\mathrm { {LC}}}^{\gamma }_{\lambda ,\eta }$
be defined as
$T_i = (C^i,w^i,l^i)$
; this correspondence is bijective, and we have
$\operatorname {\mathrm {sign}}(T) = \prod _{i=1}^{\ell (\beta )} (-1)^{\ell (\nu ^i)-1}$
and
$$\begin{align*}\operatorname{\mathrm{weight}}(T) = \sum_{i=1}^{\ell(\beta)} \operatorname{\mathrm{revmaj}}(w^i) + u(l^i) + \lvert C^i \rvert. \end{align*}$$
The thesis now follows.
7 A weight-preserving, sign-reversing involution
Our goal is to now give a positive expansion for the coefficients
$D_{\lambda ,\eta }^\gamma (q)$
. To achieve this result, we will define a weight-preserving, sign-reversing involution
$\psi \colon {\mathrm { {LC}}}_{\lambda ,\eta }^\gamma \rightarrow {\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
whose fixed points
$U^{\gamma }_{\lambda , \eta }$
give
$$\begin{align*}D_{\lambda,\eta}^\gamma (q) = \sum_{T \in U_{\lambda,\eta}^\gamma} q^{\operatorname{\mathrm{weight}}(T)}. \end{align*}$$
To construct
$\psi $
, we need to introduce a split map. Suppose
$S = (C, w, l)$
is one of the possible labeled column-composition tableaux appearing in a sequence
$T \in {\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
. Let
$C = (\alpha , c)$
, and recall that this means that the vertical bars appearing in C are in positions
$\alpha _1, \alpha _1 + \alpha _2, \dots , \alpha _1 + \dots + \alpha _{\ell -1}$
. Let
$d = \# \{ 1 \leq i \leq \alpha _1 \mid w_i < w_{i+1} \}$
, that is,
$d = \operatorname {\mathrm {asc}}(w_1, \dots , w_{\alpha _1+1})$
(we don’t count the last position if
$\alpha $
has just one part).
The idea is the following. If S has at least one bar, that is,
$\alpha \neq (\alpha _1)$
, then we set
${\mathrm { {split}}}(S) = (S_1, S_2)$
, where
$S_1$
is the portion of S occurring before the first vertical bar, and
$S_2$
is obtained from the portion of S after the first vertical bar by adding
$d + \lvert l^1 \rvert $
cells to each column,
$\lvert l^1 \rvert $
being the sum of the labels in l appearing before the first bar (see Example 7.2 for a pictorial realization). More formally, we have the following definition.
Definition 7.1. Suppose that S has at least one bar, that is,
$\alpha \neq (\alpha _1)$
. Then we say that S can split, and define 
, with
$$ \begin{align*} S_1 & = ((\alpha_1, (c_1, \dots, c_{\alpha_1})), w^1, l^1), \\ S_2 & = (((\alpha_2, \dots, \alpha_\ell), (c_{\alpha_1+1} + d + \lvert l^1 \rvert, \dots, c_{\lvert \alpha \rvert} + d + \lvert l^1 \rvert)), w^2, l^2), \end{align*} $$
where we define
$$ \begin{align*} & w^1 = (w_1, \dots, w_{\alpha_1}), \qquad \qquad \qquad & & w^2 = (w_{\alpha_1 + 1}, \dots, w_{\lvert \alpha \rvert}), \\ & l^1 = (l_1, \dots, l_{\alpha_1}), \qquad \qquad \qquad & & l^2 = (l_{\alpha_1 + 1}, \dots, l_{\lvert \alpha \rvert}). \end{align*} $$
Example 7.2. Let
$S ((\alpha , c), w, l)$
,
$\alpha =(3,1,1)$
,
$c=(1,1,1,1,2)$
,
$w=(7,4,7,7,5)$
,
$l=(0,0,2,0,3)$
. We split it after
$\alpha _1 = 3$
cells. We have one ascent in
$(7,4,7,7)$
, so
$d=1$
, and we have
$\lvert l^1 \rvert = 0+0+2 = 2$
, so we add three cells to each column in
$S_2$
, and get
Proposition 7.3. The map
${\mathrm { {split}}}$
is weight-preserving: if
${\mathrm { {split}}}(S) = (S_1, S_2)$
, then
$\operatorname {\mathrm {weight}}(S) = \operatorname {\mathrm {weight}}(S_1) + \operatorname {\mathrm {weight}}(S_2)$
.
Proof. Suppose
${\mathrm { {split}}}(S) = (S_1, S_2)$
. Let
$S = (C, w, l)$
,
$C = (\alpha , c)$
with
$\alpha _1 = v$
, and let
$\ell (C) = n$
. Let us denote
$S_1 = (C^1, w^1, l^1)$
and
$S_2 = (C^2, w^2, l^2)$
.
By definition, the weight has three components, one coming from the total size, one coming from the
$\operatorname {\mathrm {revmaj}}$
of the word w, and one coming from the labels l.
Let
$d = \operatorname {\mathrm {asc}}(w_1, \dots , w_{v+1})$
. By definition of
${\mathrm { {split}}}$
, the number of cells above
$S_1$
stays the same, while the number of cells above
$S_2$
increases by
$\ell (C_2) (d + \lvert l^1 \rvert ) = (n-v) (d + \lvert l^1 \rvert )$
, so the first component of the total weight increases by the same amount.
By definition of
$\operatorname {\mathrm {revmaj}}$
, we have
$$ \begin{align*} \operatorname{\mathrm{revmaj}}(w) & = \sum_{i \in \operatorname{\mathrm{Asc}}(w)} (n-i) \\ & = \sum_{i \in \operatorname{\mathrm{Asc}}(w^1)} (n-i) + \sum_{i \in \operatorname{\mathrm{Asc}}(w^2)} (n-(v+i)) + \chi(w_v < w_{v+1}) (n-v) \\& = \sum_{i \in \operatorname{\mathrm{Asc}}(w^1)} (n-v+v-i) + \sum_{i \in \operatorname{\mathrm{Asc}}(w^2)} (n-v-i) + \chi(w_v < w_{v+1})(n-v) \\ & = (n-v) \cdot d + \operatorname{\mathrm{revmaj}}(w^1) + \operatorname{\mathrm{revmaj}}(w^2), \end{align*} $$
so the second component of the total weight decreases by
$(n-v) \cdot d$
.
Finally, by definition of u, we have
$$ \begin{align*} u(w) & = \sum_{i=1}^n l_i \cdot (n-i) = \sum_{i=1}^v l_i \cdot (n-i) + \sum_{i=v+1}^n l_i \cdot (n-i) \\ & = \sum_{i=1}^v l_i \cdot (n-v+v-i) + \sum_{i=1}^{n-v} l_i \cdot (n-v-i) = (n-v) \lvert l^1 \rvert + u(l^1) + u(l^2), \end{align*} $$
so the third component of the total weight decreases by
$ (n-v) \lvert l^1 \rvert $
.
All these changes cancel out and so the weight is preserved, as desired.
Definition 7.4. Given two labeled column-composition tableaux
$S_1$
,
$S_2$
, we define
that is, the number of ascents in the word of
$S_1$
followed by the first letter of
$S_2$
.
Lemma 7.5. Let
$S_1, S_2$
be two labeled column-composition tableaux. There exists S, such that
${\mathrm { {split}}}(S) = (S_1, S_2)$
if and only if
$$ \begin{align} c_1(S_2) \geq c_\ell(S_1) + \operatorname{\mathrm{asc}}(S_1;S_2) + \lvert l(S_1) \rvert .\end{align} $$
If such S exists, then it is unique; we say that
$S_1$
can join
$S_2$
and set
${\mathrm { {join}}}(S_1, S_2) = S$
.
Proof. If such S exists, then (7.2) holds by construction. Suppose that (7.2) holds. Then we can define S as the labeled column composition tableau obtained by decreasing the size of each column of
$S_2$
by
$\operatorname {\mathrm {asc}}(S_1; S_2) + \lvert l(S_1) \rvert $
and then concatenating it to
$S_1$
, also concatenating their words. Equation (7.2) ensures that the result is still a column-composition tableau.
It is now immediate that
${\mathrm { {split}}}(S) = (S_1, S_2)$
and that such S is unique.
The following lemma is crucial to ensure that our sign-reversing, weight-preserving bijection is well-defined.
Lemma 7.6. Let
$S_1$
, S be labeled column-composition tableaux, and let
${\mathrm { {split}}}(S) = (S_2, S_3)$
. Then
$S_1$
can join
$S_2$
if and only if it can join S.
Proof. Since
$c_1(S_2) = c_1(S)$
, then (7.2) holds for
$S_1$
and
$S_2$
if and only if it holds for
$S_1$
and S.
We can now define our bijection as follows.
Definition 7.7. Given
$T = (T_1,\dots , T_r) \in {\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
, define
$\psi (T)$
by the following process:
-
1. if
$r=0$
, then ; -
2. if
$T_1$
can split, then ; -
3. if
$T_1$
cannot split and
$T_1$
can join
$T_2$
, then ; -
4. otherwise, we inductively define
.
;
;
;
.Theorem 7.8. Let
$$ \begin{align*} U_{\lambda,\eta}^\gamma = \Big\{ T \in {\mathrm{ {LC}}}_{\lambda,\eta}^\gamma \mid T & \text{ has no vertical bars, and for all}\ i \\ & c_1(T_{i+1}) < c_{\ell(T_i)} (T_i) + \operatorname{\mathrm{asc}}(T_i;T_{i+1}) + \lvert l(T_i) \rvert \Big\}. \end{align*} $$
Then
$$\begin{align*}D_{\lambda,\eta}^\gamma (q) = \sum_{T \in U_{\lambda,\eta}} q^{\operatorname{\mathrm{weight}}(T)}. \end{align*}$$
Proof. Since we are using the split map and its inverse, Proposition 7.3 ensures that
$\psi $
is weight-preserving. Furthermore,
${\mathrm { {split}}}$
and
${\mathrm { {join}}}$
either remove or add a single vertical bar, so
$\psi $
is sign-reversing.
We have to make sure that
$\psi $
is an involution. Let
$T = (T_1, \dots , T_r) \in {\mathrm { {LC}}}_{\lambda ,\eta }^\gamma $
. If
$\psi (T) = T$
, then clearly,
$\psi ^2(T) = T$
.
Suppose that
$\psi (T) = (T_1, \dots , T_{i-1}, {\mathrm { {split}}}(T_i), T_{i+1}, \dots , T_r)$
. Let
${\mathrm { {split}}}(T_i) = (S_1, S_2)$
. By construction,
$T_1, \dots , T_{i-1}$
cannot split, and
$T_j$
cannot join
$T_{j+1}$
for
$j < i$
. By Lemma 7.6, since
$T_{i-1}$
cannot join
$T_i$
, it also cannot join
$S_1$
. By construction,
$S_1$
and
$S_2$
can join, so
$\psi ^2(T) = T$
.
Suppose instead that
$\psi (T) = (T_1, \dots , T_{i-1}, {\mathrm { {join}}}(T_i, T_{i+1}), T_{i+2}, \dots , T_r)$
. We have that, by construction,
$T_1, \dots , T_{i-1}$
cannot split, and
$T_j$
cannot join
$T_{j+1}$
for
$j < i$
. By Lemma 7.6, since
$T_{i-1}$
cannot join
$T_i$
, it also cannot join
${\mathrm { {join}}}(T_i, T_{i+1})$
. By construction,
${\mathrm { {join}}}(T_i, T_{i+1})$
can split, so again,
$\psi ^2(T) = T$
and
$\psi $
is an involution.
The set of fixed points is the set of labeled column composition whose parts cannot split or be joined, and the conditions for that to hold are exactly the conditions given in the definition of
$U^{\gamma }_{\lambda ,\eta }$
.
Example 7.9. We can read the type of the element in Figure 6 as follows. Since the rows have lengths
$3,1,4,1$
, respectively, we know that
$\eta = (4,3,1,1)$
. Since the words in the base rows are
$(2,4,3),(1),(3,1,1,2),(2)$
, which have multiplicities given by
$1^3 2^3 3^2 4^1$
, we have
$\lambda = (3,3,2,1).$
Lastly, the labels underneath the rows rearrange to
$(3,2,2,2,1,1,0,0,0)$
, meaning
$\gamma = (3,2,2,2,1,1).$
Figure 6 A fixed point of
$\psi $
.
8 A bijection to ascent polyominoes
In this section, we define an intermediate family of objects that will turn out handy to describe our bijection between the fixed points of
$\psi $
and
$\gamma $
-parking functions, namely, ascent labeled polyominoes.
Definition 8.1. An
$m \times n$
ascent labeled parallelogram polyomino is a triple
$(P,Q,w)$
, such that
$(P,Q)$
is an
$m \times n$
parallelogram polyomino (as in Definition 2.6), and
$w \in \mathbb {N}^n$
is such that if Q has no East steps on the line
$y = i-1$
(or has only one East step if
$i=1$
, since the first step of Q must be East), then
$w_i \geq w_{i+1}$
.
Definition 8.2. Let
$(P, Q, w)$
be an ascent labeled parallelogram polyomino. Let
$\lambda \vDash n$
be the content of w (that is,
$m(w) = 1^{\lambda _1} 2^{\lambda _2}\cdots $
), and let
$\eta \vdash n$
be the partition whose block sizes are the lengths of the maximal streaks of North steps in P, in some order. Let
and let
$\gamma $
be the partition obtained by rearranging
$(\beta _1, \dots , \beta _n)$
and removing zeros.
We define
${\mathrm { {type}}}(P, Q, w) = (\lambda , \eta , \gamma )$
, and call
${\mathcal {P}}_{\lambda , \eta }^\gamma $
the set of ascent labeled polyominoes of type
$(\lambda , \eta , \gamma )$
. Note that the height is fixed by the type but the width isn’t, as it depends on the number of ascents in w.
We will now give a bijection
$\varphi \colon U^{\gamma }_{\lambda ,\eta } \rightarrow {\mathcal {P}}^\gamma _{\lambda ,\eta }$
from the set of fixed points of
$\psi $
of given type, and ascent labeled parallelogram polyominoes of the same type.
In order to describe the bijection, for
$T = (T_1, \dots , T_r)\in U^{\gamma }_{\lambda ,\eta }$
, we need to define a triple
$\varphi (T) = (P(T),Q(T),w(T))$
corresponding to the polyomino and its labels.
Definition 8.3. Let
$T = (T_1, \dots , T_r)\in U^{\gamma }_{\lambda ,\eta }$
, with
$T_i = (C^i, w^i, l^i)$
. First, we define 
(the concatenation).
Next, let
$l(T) = l^1 \cdots l^r$
and let
$r_i(T) = l(T)_i + \chi (i \in \operatorname {\mathrm {Asc}}(w))$
. We define
that is, the path with
$r_i(T)$
East step on the line
$y=i-1$
, plus
$1$
if
$i=1$
.
Finally, let
$s_i(T) = c_{\ell (T_i)}(T_i) + \operatorname {\mathrm {asc}}(T_i; T_{i+1}) + \lvert l^i \rvert - c_1(T_{i+1})$
, which is guaranteed to be positive by the fact that T is a fixed point of
$\psi $
. We set
Example 8.4. We demonstrate the bijection for the sequence of labeled column composition tableaux
$T \in U^{(3,2,2,2,1)}_{(3,3,1,1),(4,3,1,1)}$
appearing in Figure 6. We have
$w = w(T) = (2,4,3,1,3,1,1,2,2)$
and
$l = l(T) = (0,2,0,2,2,1,1,0,3)$
.
We have
$\operatorname {\mathrm {Asc}}(w) = \{1, 4, 7\}$
, so we get
$$\begin{align*}Q(T) = E E^{0+1} N E^{2+0} N E^{0+0} N E^{2+1} N E^{2+0} N E^{1+0} N E^{1+1} N E^{0+0} N E^{3+0} N. \end{align*}$$
Finally, we have
$s_1(T) = 0 + 1 + 2 - 2 = 1$
,
$s_2(T) = 2 + 1 + 2 - 0 = 5$
,
$s_3(T) = 0 + 1 + 4 - 2 = 3$
, and
$s_4(T) = 2 + 0 + 3 - 0 = 5$
, so we end up with
$$\begin{align*}P(T) = N^3 E^1 N^1 E^5 N^4 E^3 N^1 E^5 E. \end{align*}$$
We refer to Figure 7 for the image of T under
$\varphi $
. The North steps of
$P(T)$
are labeled with the word w written from bottom to top. The North segments are also distinguished in our picture with a red line on its left. We see that the vertical segments of
$P(T)$
have lengths
$3,1,4,1$
, which rearranges to
$\eta = (4,3,1,1)$
. The green segments highlight the horizontal segments of
$Q(T)$
(ignoring the first, mandatory East step), and along each horizontal line, we have lengths
$(1,2,0,3,2,1,2,0,3)$
. Since
$\operatorname {\mathrm {Asc}}(w) = \{1,4,7\}$
, we subtract term by term to see that
$$\begin{align*}(1,2,0,3,2,1,2,0,3) - (1,0,0,1,0,0,1,0,0)= (0,2,0,2,2,1,1,0,3) \end{align*}$$
has nonzero parts that rearrange to
$\gamma = (3,2,2,2,1,1)$
, as expected.
Figure 7 The image
$\varphi (T)$
of the fixed point in Figure 6.
Theorem 8.5. The map
$\varphi (T) = (P(T),Q(T),w(T))$
is a bijection between
$U_{\lambda ,\eta }^\gamma $
and
${\mathcal {P}}_{\lambda ,\eta }^\gamma $
, such that
$\operatorname {\mathrm {weight}}(T) = \operatorname {\mathrm {area}}(\varphi (T))$
, that is,
$\varphi $
is weight-preserving.
Proof. We describe the inverse
$\varphi ^{-1}$
instead. Starting with
$S = (P,Q,w) \in {\mathcal {P}}_{\lambda , \eta }^\gamma $
, for each maximal vertical segment in P, draw the maximal rectangle contained in S that has that streak as one of the sides and whose perimeter does not contain any East step in Q. Then slide all of the labels in these vertical segments to the opposite side of the maximal rectangle (see Step 2 of Figure 8).
Figure 8 A pictorial description of
$\varphi ^{-1}$
.
To construct
$(T_1,\dots , T_{\ell (\lambda )})$
, we can determine the tableaux by looking at the rectangles: if
$T_i = (C^i, w^i, l^i)$
, then
$C^i$
is the column-composition tableau obtained by rotating the rectangle delimited by the
$i^{\textit {h}}$
vertical segment in S;
$w^i$
is the sequence of labels appearing in the rectangle, read from bottom to top; if
$l = l^1 \cdots l^{\ell (\lambda )}$
, then
$l_j + \chi (j \in \operatorname {\mathrm {Asc}}(w))$
is the number of cells of S in the
$j^{\textit {h}}$
row that are outside the maximal rectangle and whose bottom segment is an East step of Q (since the condition on P forces
$l_j \geq 0$
, so if
$j \in \operatorname {\mathrm {Asc}}(w)$
, then there must be at least one such cell). This is better seen by rotating S by
$90$
degrees clockwise (as in Figure 8, Step 3). Let us call this rotated picture
$S'$
.
We should note that since
$(P,Q,w)$
gives a parallelogram polyomino, the
$i+1^{\textit {h}}$
vertical segment in P occurs strictly right of the
$i^{\textit {h}}$
vertical segment and also strictly left of Q. Using the translation to
$T = \varphi ^{-1}(S)$
, this is equivalent to say that
$$\begin{align*}c_1(T_{i+1}) < c_{\ell(T_i)}(T_{i}) + \sum_{j =1}^{\ell(T_i)} \chi(j \in \operatorname{\mathrm{Asc}}(w^i w^{i+1})) + \sum_{j=1}^{\ell(T_i)} l_i \end{align*}$$
or rather
$$\begin{align*}c_1(T_{i+1}) < c_{\ell(T_i)}(T_{i}) + \operatorname{\mathrm{asc}}(T_i;T_{i+1}) + \lvert l(T_i)\rvert. \end{align*}$$
Since the defining property making P a path above Q with respect to w converts directly to the defining relation for elements
$T \in U_{\lambda ,\eta }^\gamma $
, then
$\varphi $
is a bijection.
We are now going to show that
$\varphi $
is weight-preserving. First, draw a path along the cells of
$S'$
starting from the top left cell, and moving East if there are labels directly East, and moving South otherwise (see Figure 9). Now the area can be computed by counting the number of unit cells with no dashed line through it, those cells covering the minimal area, we remove as normalization. The area above the dashed line equals the weight contribution in the
$T_i$
given by the cells above the base row.
Figure 9
$\varphi $
is weight-preserving.
The cells below the dashed line will be counted in the following way: In
$S'$
, directly South of the label
$w_j$
, there are by construction
$l_j + \chi (j \in \operatorname {\mathrm {Asc}}(w))$
cells which are adjacent to the path on the left. Each of these cells have the same number of cells weakly East of them, namely, the number of cells between
$w_j$
and the next base row. The number of cells in these rows is then
$$\begin{align*}(l_j + \chi(j \in \operatorname{\mathrm{Asc}}(w))) \times \# \{ \text{cells in the same base and on the right of}\ w_j \}. \end{align*}$$
Taking the sum over all j, we get the number of cells below the dashed line. But now the second and the third factor contributing to
$\operatorname {\mathrm {weight}}(T)$
are exactly
$$\begin{align*}\sum_{j \in \operatorname{\mathrm{Asc}}(w)} \# \{ \text{cells in the same base and on the right of}\ w_j \} \end{align*}$$
and
$$\begin{align*}\sum_j l_j \times \# \{ \text{cells in the same base and on the right of}\ w_j \} \end{align*}$$
(as we described in Example 6.4).
Finally, it is immediate to see that the number of cells on the dashed blue line is exactly
$m+n-1$
. So, by Definition 2.7, indeed,
$\operatorname {\mathrm {weight}}(T) = \operatorname {\mathrm {area}}(\varphi (T))$
, as desired.
In the end, we get the following result.
Theorem 8.6.