The symmetric functions catalog

An overview of symmetric functions and related topics

2024-01-19

For background, see the canonical sources by J. Haglund [Hag07], and F. Bergeron[Ber09]. However, there has been a lot of exciting new development since the writing of these books.

Introduction

The ring of diagonal harmonics is defined as the set of polynomials in $\setC[X,Y],$ such that for all $a+b \ge 0$ following partial differential equations hold:

\[ \sum_{1\leq j \leq n} \partial^{a}_{x_j} \partial^{b}_{y_j} f(X,Y) = 0. \]

We have that $\symS_n$ act diagonally by permuting the variables in each alphabet, making the space of diagonal harmonics into a bigraded $\symS_n$-module $\mathcal{DH}_n.$ One can show that $\mathcal{DH}_n$ is also the ring of diagonal coinvariants

\[ \mathcal{DH}_n = \setC[X,Y]/I, \quad I = (\xvec,\yvec) \cap \setC[X,Y]^{\symS_n}. \]

A result by M. Haiman [Hai94] states that the bigraded Frobenius characteristic of $\mathcal{DH}_n$ is given by $\nabla \elementaryE_n.$ The Shuffle theorem gives a combinatorial expression for $\nabla \elementaryE_n.$

Definitions

Given a partition $\mu,$ we define

\[ B(\mu) \coloneqq \sum_{\square \in \mu} q^{\arm'(\square)} t^{\leg'(\square)}. \]

Thus, a cell in row $i,$ column $j$ contributes with $q^{j-1} t^{i-1}.$

Example (Example $B(331)$).

We have that $B(\mu) = 1+q+q^2+q^3 + t(1+q+q^2+q^3)+t^2,$ as the monomials arise from the following Young diagram:

$1$$q$$q^2$$q^3$
$t$$tq$$q^2t$$q^3t$
$t^2$   

Let $f$ be any symmetric function. We define an operators $\Delta_f$ and $\Delta'_f$ on $\spaceSym(q,t),$ by setting

\[ \Delta_f( \macdonaldH_\mu ) \coloneqq f[ B(\mu) ] \cdot \macdonaldH_\mu \text{ and } \Delta'_f( \macdonaldH_\mu ) \coloneqq f[ B(\mu) - 1]\cdot \macdonaldH_\mu. \]

Note the use of plethystic substitution. Hence, the modified Macdonald polynomials are by definition eigenfunctions of $\Delta_f.$ Moreover, we define $\nabla$ as the special case

\[ \nabla( \macdonaldH_\mu ) \coloneqq \Delta_{\elementaryE_n} = \prod_{(i,j) \in \mu} ( q^{i-1} t^{j-1} ) \cdot \macdonaldH_\mu. \]

In other words,

\[ \nabla( \macdonaldH_\mu ) = q^{\partitionN(\mu)} t^{\partitionN(\mu')} \macdonaldH_\mu \]

Relations among the operators

In, [HRW18] it is stated that

\[ \Delta_{\elementaryE_k} \elementaryE_n = \Delta'_{\elementaryE_k} \elementaryE_n + \Delta'_{\elementaryE_{k-1}} \elementaryE_n. \]

$\mathcal{DH}_n$ and the shuffle theorem

Properties of $\mathcal{DH}_n$

Let $DH_n(q,t)$ be the bigraded Frobenius characteristic of $\mathcal{DH}_n.$ As shown by M. Haimain [Hai94], $DH_n(q,t) = \nabla \elementaryE_n.$

The (total) dimension of $\mathcal{DH}_n$ is $(n+1)^{n-1}.$ This is sequence A000272 and in particular, is the number of parking functions of size $n.$ The dimension can be refined by degree — the bigraded Hilbert series is given by

\[ \langle DH_n(q,t), \completeH_{1^n} \rangle = \sum_{w \in PF(n)} q^{\dinv(w)} t^{\area(w)}. \]

It is an open problem to give a combinatorial argument why this sum is symmetric in $q$ and $t.$

It is a major open problem to find a combinatorial expression for the Schur expansion of $DH_n(q,t)$ (see [Prob. 2.15, Hag07]). However, a combinatorial formula for the LLT polynomials would solve this problem as well.

Example (Example of $\nabla \elementaryE_n$ in the Schur basis).

For $n=1,2,3$ we have the following Schur expansions

$n$$\nabla \elementaryE_n$
$1$$\schurS_{1}$
$2$$\schurS_{2} + (q+t) \schurS_{11}$
$3$$\schurS_{3}+(q^2+t^2+qt+q+t) \schurS_{21} + (q^3+t^3+q^2 t + qt^2 + qt) \schurS_{111}$

Some special values of $DH_n(q,t)$ are given as \[ DH_n(q,0) = [n]_q!, \qquad q^{\binom{n}{2}}DH_n(q,q^{-1}) = ([n+1]_q)^{n-1} \]

The latter formula was originally conjectured by R. Stanley.

Also, by summing over spanning trees of $\{0,1,\dotsc,n\},$ and weighting them by inversions, we have

\[ DH_n(q,1) = \sum_{T} q^{\inv(T)}. \]

The $qt$-Catalan numbers show up here by computing the sign representation:

\[ \langle DH_n(q,t), \elementaryE_{n} \rangle = \sum_{ P \in \DP(n) } q^{\dinv(P)} t^{\area(P)} = \sum_{ P \in \DP(n) } q^{\area(P)} t^{\bounce(P)}. \]

These identities were proved in [Thm. I.2, GH02].

M. Haiman proved in [Thm. 3.10, Hai02b] a connection with Macdonald polynomials, among many other things.

Theorem (Haiman (2002)).

Set $T_\mu \coloneqq t^{n(\mu)} q^{n(\mu')},$ $M\coloneqq(1-q)(1-t),$ and

\[ w_\mu \coloneqq \prod_{\square \in \mu} (q^{\arm(\square)} - t^{\leg(\square)+1}) (t^{\leg(\square)}- q^{\arm(\square)+1} ), \qquad \Pi_\mu \coloneqq \prod_{\square \in \mu/(1)} \left(1- q^{\arm'(\square)} t^{\leg'(\square)} \right). \]

Then, with $\macdonaldH_\mu$ being the modified Macdonald polynomials,

\[ DH_n(q,t) = \sum_{\mu \vdash n} \frac{ T_\mu M \macdonaldH_\mu \Pi_\mu \elementaryE_n[B(\mu)] }{ w_\mu }. \]

The shuffle theorem

The shuffle conjecture was originally stated in [HHLRU05]. It was later refined to what is known as the compositional shuffle conjecture, presented in [HMZ12]. Both these conjectures have now been settled.

Theorem (Carlsson–Mellit [CM17]).

We have the identity

\[ \nabla \elementaryE_n = \sum_{w \in WPF(n)} t^{\area(F)}q^{\dinv(F)} \xvec_w \]

where the sum is taken over so-called word parking functions.

After using the zeta map, as defined in [p. 82, Hag07], the right hand side can be expressed using vertical-strip LLT polynomials as

\[ \nabla \elementaryE_n = \sum_{\avec \in \DP(n)} t^{\bounce(\avec)} \LLT_{\avec,\svec}(\xvec;q) \]

where $\svec$ marks every corner as strict, see [AP18] for definitions.

In fact, they prove a stronger statement, the compositional shuffle theorem.

Theorem (Carlsson–Mellit [CM17]).

Let $C_a$ be the operator defined as

\[ C_a f[X] \coloneqq [z^a] (-1/q)^{a-1} f[X-(1-q^{-1})/z] \sum_{z\geq 0} z^m \completeH_m[X]. \]

We then have the identity

\[ \nabla (C_{\alpha_1} C_{\alpha_2}\dotsm C_{\alpha_\ell} 1) = \sum_{\substack{w \in WPF(n) \\ touch(w)=\alpha }} t^{\area(F)}q^{\dinv(F)} \xvec_w \]

where the sum is taken over word parking functions with $\alpha$ specifying the touch-points.

For more background and references on the Shuffle theorem see S. van Willigenburg's survey, http://de.arxiv.org/pdf/1905.06970.pdf.

A generalized shuffle theorem

A generalization of the Shuffle theorem is given in [BHMPS21a], where on the combinatorial side, the sum now ranges over lattice path under any line (not just a diagonal).

qt-Narayana numbers

The $qt$-Narayana numbers are described in [ADDHB14]. They may be defined as

\[ N_{a,b}(q,t) = \sum_{P \in Polyo(b,a+1-b)} q^{\area(P)}t^{\bounce(P)} \]

where the sum is taken over certain pairs of non-intersecting paths confined to a $b \times (a+1-b)$-rectangle (so called parallelogram polyominoes).

The authors show in [Thm. 6.1, ADDHB14] that

\[ N_{a,b}(q,t) = (qt)^{a} \langle \nabla \elementaryE_{a-1}, \completeH_{b-1} \completeH_{a-b} \rangle. \]

Rational Shuffle theorem

A bigraded deformation of the $q,t$-Catalan numbers are introduced in [GM13]. The rational shuffle conjecture is then stated in [Eq. 51, GN15] and independently in [Hik14] and [Arm12].

The extended rational shuffle conjecture is first stated in [BGLX15]. They introduce certain operators $Q_{m,n},$ generalizing operators $\widetilde{P}_{m,n}$ by Gorsky and Mazin (which required that $m,n$ be comprime), and conjecture a combinatorial formula for the operators when applied to $(-1)^n.$

Note that $Q_{n+1,n} (-1)^n = \nabla \elementaryE_n,$ so we have the shuffle theorem as a special case.

Using notation similar to that of [Eq. (11.97), Hag15], and [QR18], we set, for coprime $m,n,$

\[ H_{(m,n)(\xvec;q,t)} \coloneqq \sum_{(m,n) \text{ parking functions } P} q^{\area(P)} t^{\dinv(P)} \gessel_{n,\DES(P)}(\xvec). \]

The functions $H_{(m,n)}(\xvec;q,t)$ are also known as Hikita polynomials (for coprime $m,n$), after the introduction in [Hik14]. Bergeron, Garsia, Leven and Xin then introduce the extended Hikita polynomials,

\[ H_{(km,kn)(\xvec;q,t)} \coloneqq \sum_{(km,kn) \text{ parking functions } P} [ret(P)]_{1/t} q^{\area(P)} t^{\dinv(P)} \gessel_{n,\DES(P)}(\xvec). \]

where $ret(P)$ is the smallest integer $j \gt 0$ such that the supporting path of $P$ goes through $(jm,jn).$ One can then show (see [Eq. 6.3, BGLX15]) that $H_{(m,n)}$ is a positive linear combination of vertical-strip LLT polynomials. In [KK18], a close relationship between Hikita polynomials and Catalan symmetric functions is studied.

The operators $Q_{m,n}$ are defined recursively via the Bergeron–Garsia operators $D_k$ (acting on $F \in \spaceSym$)

\[ D_k F[X] \coloneqq [z]^{k} F\left[ X + \frac{(1-q)(1-t)}{z} \right] \sum_{j \geq 0} (-z)^j \elementaryE_j[X]. \]

The construction of the $Q_{m,n}$ is quite involved. Moreover, it seems that different authors use slightly different definitions which leads to a sign difference.

It is shown by [GN15] that for coprime $m,n,$ and in [QR18] that

\[ \nabla Q_{m,n} \nabla^{-1} = Q_{m+n,n} \qquad \nabla Q_{kn,n} \nabla^{-1} = Q_{(k+1)n,n}. \]

A. Mellit proved the following theorem in [Mel16].

Theorem (The (extended) rational shuffle theorem).

For coprime $m,n$ and $k \ge 0,$ we have that

\[ H_{(km,kn)}(\xvec;q,t) = Q_{km,kn} (-1)^n. \]

In fact, he proves the compositional refinement stated in [Conj. 3.3, BGLX15].

In [Hik14], it is shown that $B_{m,n}(\xvec;q,t)$ is the Frobenius character of the action in the homology of a certain Springer fiber in the affine flag variety equipped with a certain filtration.

The rational Catalan numbers are defined as $\catalan_{a/b} = \frac{1}{a+b}\binom{a+b}{a,b}$ for coprime $a,b.$ We have that for coprime $m,n$ that

\[ \langle H_{(m,n)}(\xvec;q,t), \elementaryE_n \rangle_* = \catalan_{m/n}. \]

where $\langle \cdot, \cdot \rangle_*$ is a certain deformation of the Hall innter product, see [Eq.(4.6) and (6.29), BGLX15].

An expression for $H_{(m,n)}(\xvec;q,t)$ can also be given by using so called Tesler matrices.

See [QR18] for recent results on Schur coefficients of $H_{(m,n)}(\xvec;q,t).$

The square paths theorem, $\nabla \powerSum_n$

The square path conjecture was formulated by N. Loehr and G. Warrington [LW07b]. It provides a combinatorial expression for $\nabla \powerSum_n$ as a sum over preference function.

A preference functions is a map $f:[n] \to [n].$ A parking function is any preference function such that $|f^{-1}([k])| \geq k$ for all $k \in [n].$ With the parking language, we think of $f(i)$ as the preference of car $i.$ There is a way to associate a lattice path to any preference function as follows. In an $n\times n$-square, write the numbers $f^{-1}(i)$ in column $i,$ starting from $i=1.$ The numbers in each column are increasing upwards, and the $j$th number written is placed in row $j.$ Then there is a unique lattice path from $(0,0)$ to $(n,n)$ using North and East steps, such that the numbers are written immediately to the right of the North steps.

Example (Path from preference function).

For example, we have that the preference function $(1,5,1,2,1)$ give rise to the path $P =\mathtt{nnneneene},$ and the diagram

    $2$
 $4$   
$5$    
$3$    
$1$    

The area of a preference function is the number of full cells above the lowest diagonal which contains a car. Thus, if the path is a Dyck path (and the preference function is a parking function), the area coincide with the classical notion of area. The dinv statistic is similar to the inv-statistic for LLT polynomials, but one must add the number of cars strictly below the $x=y$ diagonal.

E. Sergel [Ser17] gave a proof of the square path conjecture.

Theorem (E. Sergel, (2017)).

We have the expansion in the fundamental quasisymmetric functions

\[ (-1)^{n-1} \nabla \powerSum_n = \sum_{w \in Pref(n)} t^{\area(w)} q^{\dinv(w)} \gessel_{\mathrm{IDES}(w)}(\xvec), \]

where the sum is taken over all preference functions of size $n.$

It is rather straightforward to express this as a sum over vertical-strip LLT polynomials. A path arising from a preference function of size $n,$ can be described using a weak composition $\alpha,$ such that $\alpha_i \coloneqq |f^{-1}(i)|.$ It is then rather straightforward how to define $\area(\alpha)$ and $\mathrm{carsBelow}(\alpha),$ the number of cars below the main diagonal. Finally, we can associate an $n$-tuple of skew shapes, $\nuvec,$ determined by the positions of the cars. We therefore get the following:

Proposition.

We have that

\begin{equation*} (-1)^{n-1} \nabla \powerSum_n = \sum{\alpha} t^{\area(\alpha)} q^{\mathrm{carsBelow}(\alpha)} \LLT_{\nuvec(\alpha)}(\xvec;q) \end{equation*}

where the sum ranges over all weak compositions of length $n$ and size $n.$

Example (Schur-expansion of $(-1)^{n-1} \nabla \powerSum_n$).
$n$$(-1)^{n-1} \nabla \powerSum_n$
$1$$\schurS_{1}$
$2$$\schurS_{2} + (q + t + q t) \schurS_{11}$
$3$$\schurS_{3} + (q + q^2 + t + q t + q^2 t + t^2 + q t^2 + q^2 t^2)\schurS_{21} +$
 $+(q^3 + q t + q^2 t + q^3 t + q t^2 + q^2 t^2 + q^3 t^2 + t^3 + q t^3 + q^2 t^3) \schurS_{111}$

The Delta conjecture, $\Delta'_{\elementaryE_k} \elementaryE_n$

The Delta conjecture generalizes the Shuffle theorem, and was stated by J. Haglund, J. Remmel, A. Wilson [HRW18].

There are two versions of the Delta conjecture, the rise version and the valley version:

Conjecture (Delta conjecture, J. Haglund, J. Remmel, A. Wilson).

The rise version:

\[ \Delta'_{\elementaryE_k} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Rise(w)} \left(1 + z/t^{a_i(w)}\right) \]

The valley version:

\[ \Delta'_{\elementaryE_k} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Val(w)} \left(1 + z/t^{d_i(w)+1}\right) \]

The rise version has a compositional generalization, and this has recently been proved in https://arxiv.org/pdf/2011.11467.pdf.

The generalized Delta conjecture, $\Delta_{\completeH_m} \Delta'_{\elementaryE_{n-k-1}} \elementaryE_n$

Also Conj 4.2, 4.3 https://arxiv.org/pdf/2003.12048.pdf

The extended Delta conjecture, $\Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n$

J. Haglund, J. Remmel, A. Wilson [HRW18] also formulated the extended Delta conjecture, which gives a combinatorial formula for

\[ \Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n, \quad k \lt n. \]

D. Qiu and A. Wilson [QW] gave the valley version of the extended Delta conjecture.

Conjecture (Extended Delta conjecture, J. Haglund, J. Remmel, A. Wilson).

The rise version:

\[ \Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n,r)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Rise(w)} \left(1 + z/t^{a_i(w)}\right) \]

Valley version:

\[ \Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n,r)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Val(w)} \left(1 + z/t^{d_i(w)+1}\right) \]

The valley version has been proved true for $q=0$ or $t=0.$

See also [(6.10), Ber20a]. For some recent results see [DI].

Generalized Delta square conjecture

The Generalized delta square conjecture (stated in [DIW]) is about a combinatorial interpretation of

\[ \frac{[n-k]_t}{[n]_t} \Delta_{\completeH_m} \Delta_{\elementaryE_{n-k}} \omega(\powerSum_n). \]

Remark 3.14 shows that this (if the conjecture is true) can be expressed as a sum of LLT polynomials.

See conj. 4.5, 4.6: Valley version stated in [IW].

Some questions solved in [Rom22].

Conjecture on $\nabla^m \schurS_\lambda$

N. Loehr and G. Warrington [LW10] conjectured a combinatorial expansion of $\nabla^m \schurS_\lambda.$

This conjecture (in fact, a much stronger result) was proved by J. Blasiak, M. Haiman, J. Morse, A. Pun and G. Seelinger, in [BHMPS21bBHMPS21c]. Their work also gives a new proof of the Shuffle theorem and similar other identities.

Conjecture on $\nabla \monomial_\lambda$

Conjecture (Bergeron–Garsia–Haiman–Tesler (1999), [Conj. IV, BGHT99]).

We have for every pair of partitions $\lambda,$ $\mu,$

\[ (-1)^{|\lambda|-\length(\lambda)} \langle \nabla \monomial_\lambda, \schurS_\mu \rangle \in \setN[q,t]. \]

This also appears in [Conj. 11, LW07b].

Other conjectures

Conjecture (Bergeron–Garsia–Haiman–Tesler (1999),[Conj. I, BGHT99]).

We have for every pair of partitions $\lambda,$ $\mu$ and positive integer $m,$

\[ (-1)^{\iota(\lambda)} \langle \nabla^m \schurS_\lambda, \schurS_\mu \rangle \in \setN[q,t], \]

where

\[ \iota(\lambda) \coloneqq \binom{\length(\lambda)}{2} + \sum_{\lambda_i < (i-1)} (i-1-\lambda_i). \]
Conjecture (Bergeron–Garsia–Haiman–Tesler (1999), [Conj. II, BGHT99]).

We have for every pair of partitions $\lambda,$ $\mu,$

\[ (-1)^{|\lambda|-\length(\lambda)} \langle \nabla \macdonaldH_\lambda(\xvec;0,t), \schurS_\mu \rangle \in \setN[q,t], \]

where

\[ \iota(\lambda) \coloneqq \binom{\length(\lambda)}{2} + \sum_{\lambda_i < (i-1)} (i-1-\lambda_i). \]

In [BGHT99] and later [Conj. 3.20, Hai02b], the following conjecture is stated:

Conjecture (Bergeron–Garsia–Haiman–Tesler (1999)).

The expression

\[ \nabla_{\schurS_\nu} \elementaryE_n \]

is $(q,t)$-Schur positive polynomial for all $\nu$ and $n.$

List of papers and preprints

References