# Comptes Rendus
Mathématique

Damir Yeliussizov

Dual Grothendieck polynomials via last-passage percolation

Volume 358, issue 4 (2020), p. 497-503

Published online: 28 July 2020

https://doi.org/10.5802/crmath.67

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/

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Combinatorics, Probability Theory / Combinatoire, Théorie des probabilités

# Dual Grothendieck polynomials via last-passage percolation

## Polynômes de Grothendieck duales par percolation de dernier passage

Damir Yeliussizov<sup>a</sup>

<sup>a</sup> Kazakh-British Technical University, 59 Tole bi st, Almaty 050000, Kazakhstan.

E-mail: yeldamir@gmail.com.

**Abstract.** The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous K-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.

**Résumé.** L'anneau de fonctions symétriques a une base de polynômes de Grothendieck duales qui sont des déformations K-théoriques non homogènes des polynômes de Schur. Nous prouvons que les polynômes de Grothendieck duales déterminent distributions des colonnes pour un modèle de percolation dirigée de dernier passage.

**Mathematical subject classification** (2010). 05E05, 60K35, 60C05.

**Funding.** The author was partially supported by MSERK grants AP08053036, AP05131179.

Manuscript received 25th February 2020, revised 1st May 2020, accepted 3rd May 2020.

## 1. Introduction

In this note we show a surprising connection between (i) the dual Grothendieck polynomials that are deformations of Schur polynomials arising in K-theoretic Schubert calculus, and (ii) a directed last-passage percolation model (which can also be viewed as the corner growth model or a totally asymmetric simple exclusion process).

### 1.1. Dual Grothendieck polynomials

The ring of symmetric functions has an inhomogeneous basis {$g_\lambda$} called the dual Grothendieck polynomials. The symmetric polynomials $g_\lambda(x_1, \dots, x_n)$ can be defined via the following combinatorial formula

$$g_{\lambda}(x_1, \dots, x_n) := \sum_{\pi: \mathrm{sh}(\pi) = \lambda} \prod_{i=1}^{n} x_i^{c_i(\pi)},$$
where the sum runs over *plane partitions* $\pi$ of shape $\lambda$ with largest entry at most $n$ and $c_i(\pi)$ is the number of columns of $\pi$ containing $i$. It is easy to see that $g_\lambda = s_\lambda +$ lower degree terms, where $s_\lambda$ is the Schur polynomial. This basis was explicitly introduced and studied in [7] (and earlier implicitly in [3, 8]) in relation to the K-theory of Grassmannians. More properties of these functions can also be found in [13, 16].

## 1.2. Directed last-passage percolation

Let $W = (w_{ij})_{i,j \ge 1}$ be a random matrix with independent entries $w_{ij}$ that have geometric distribution with parameters $q_j \in (0, 1)$, i.e.

$$ \mathrm{Prob}(w_{ij} = k) = (1 - q_j) q_j^k, \quad k \in \mathbb{N}. $$

A lattice path $\Pi$ with vertices in $\mathbb{N}^2$ is called a *directed path* if it uses only steps of the form $(i, j) \to (i+1, j)$, $(i, j+1)$. Define the *last-passage times* $G(m, n)$ as follows:

$$ G(m, n) := \max_{\Pi: (1,1) \to (m,n)} \sum_{(i,j) \in \Pi} w_{ij}, $$

where the maximum is over directed paths $\Pi$ from (1, 1) to (m, n). The function G presents certain random growth. This probabilistic model, which can also be viewed as the corner growth model or a totally asymmetric simple exclusion process (TASEP), was studied intensively (especially in the i.i.d. case $q_j = q$), see [1, 4, 6, 10, 11] and references therein. Let us call the matrix $G = (G(m, n))_{m,n \ge 1}$ as the *percolation matrix*.

## 1.3. Column distributions of the percolation matrix

Our main result is the formula showing that joint distribution of elements along any column in the percolation matrix $G$ is proportional to evaluations of dual Grothendieck polynomials. Let

$$ \mathcal{P}_m := \{\lambda = (\lambda_1, \dots, \lambda_m) \in \mathbb{N}^m : \lambda_1 \ge \dots \ge \lambda_m \ge 0\} $$

be the set of integer partitions with at most $m$ parts.

**Theorem 1.** Let $\lambda = (\lambda_1, \dots, \lambda_m) \in \mathcal{P}_m$. The following formula holds

$$ \mathrm{Prob}(G(m, n) = \lambda_1, \dots, G(1, n) = \lambda_m) = g_{\lambda}(q_1, \dots, q_n) \prod_{i=1}^{n} (1 - q_i)^m $$

On one hand, this formula can be viewed as a natural probabilistic interpretation of dual Grothendieck polynomials. On the other hand, it can also be used for computing distribution formulas in the percolation matrix. We prove Theorem 1 combinatorially, using certain bijection between plane partitions and integer matrices. We then give some applications. For example, we present new generating function identities for dual Grothendieck polynomials and determinantal formulas for distributions of the percolation matrix.

# 2. Proof of the main theorem

## 2.1. Plane partitions and $\mathbb{N}$-matrices

An $\mathbb{N}$-matrix is a matrix of nonnegative integers with only finitely many nonzero elements. A *plane partition* is an $\mathbb{N}$-matrix $\pi = (\pi_{ij})_{i,j \ge 1}$ such that

$$ \pi_{ij} \ge \pi_{i+1,j}, \quad \pi_{ij} \ge \pi_{i,j+1}, \quad i, j \ge 1. $$

The shape of $\pi$ is defined as $\mathrm{sh}(\pi) := \{(i, j) : \pi_{ij} > 0\}$.
Given a plane partition $\pi$, define the descent level sets

$$D_{i\ell} := \{j : \pi_{ij} = \ell > \pi_{i+1j}\},$$

i.e. $D_{i\ell}$ is the set of column indices of the entry $\ell$ in $i$th row of $\pi$ that are strictly larger than the entry below. Let $d_{i\ell} := |D_{i\ell}|$ and $D := (d_{i\ell})_{i,\ell \ge 1}$.

Define the map $\Phi: \{\text{plane partitions}\} \to \{\mathbb{N}-\text{matrices}\}$ by setting

$$\Phi(\pi) = D.$$

For example,

$$\Phi: \begin{bmatrix} 4 & 4 & 2 \\ 4 & 2 & 1 \\ 2 & 2 \end{bmatrix} \to \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 2 & 0 & 0 \end{pmatrix}$$

In fact, $\Phi$ is a bijection; we can uniquely reconstruct $\pi$ given the matrix $D$. We refer to [14, 15] for more on this bijection. Denote by PP$(m, n)$ the set of plane partitions with at most $m$ rows and largest entry at most $n$. In particular, if $\pi \in \text{PP}(m, n)$, then $\text{sh}(\pi) \in \mathcal{P}_m$ and $D = \Phi(\pi)$ has at most $m$ rows and $n$ columns that are nonzero.

**Lemma 2.** Let $W = (w_{i\ell})_{i,\ell=1}^{m,n}$ be an $m \times n$ matrix, where $w_{i\ell}$ are independent geometrically distributed random variables with parameters $q_\ell$. Let $\pi \in \text{PP}(m, n)$. Then

$$\text{Prob}(W = \Phi(\pi)) = \prod_{\ell=1}^{n} (1 - q_{\ell})^{m} q_{\ell}^{c_{\ell}(\pi)}$$

where $c_\ell(\pi)$ is the number of columns of $\pi$ containing $\ell$.

**Proof.** Let $(d_{i\ell}) = \Phi(\pi)$, i.e. $d_{i\ell} = |\{j : \pi_{ij} = \ell > \pi_{i+1j}\}|$ for $i \in [1, m], \ell \in [1, n]$. Since the entries of $W$ are independent we obtain that

$$\text{Prob}(W = \Phi(\pi)) = \prod_{i=1}^{m} \prod_{\ell=1}^{n} \text{Prob}(w_{i\ell} = d_{i\ell}) = \prod_{i=1}^{m} \prod_{\ell=1}^{n} (1 - q_{\ell}) q_{\ell}^{d_{i\ell}} = \prod_{\ell=1}^{n} (1 - q_{\ell})^{m} q_{\ell}^{c_{\ell}(\pi)}$$

as $\sum_i d_{i\ell} = c_\ell(\pi)$, as needed. $\square$

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**Lemma 3.** Let $\pi \in \text{PP}(m, n)$ and $\Phi(\pi) = D = (d_{i\ell})$. Let $\lambda = \text{sh}(\pi)$ be the shape of $\pi$. We have for all $k \in [1, m]$

$$\lambda_k = \max_{\Pi:(k,1) \to (m,n)} \sum_{(i,\ell) \in \Pi} d_{i\ell}, \quad (1)$$

where the maximum is over directed paths $\Pi$ from $(k, 1)$ to $(m, n)$.

**Proof.** Take an arbitrary directed path $\Pi$ from $(k, 1)$ to $(m, n)$. Then the descent level sets $D_{i\ell}$ for $(i, \ell) \in \Pi$ are pairwisely disjoint. Using this property and since $i \ge k$ for all $(i, \ell) \in \Pi$, we obtain

$$\sum_{(i, \ell) \in \Pi} d_{i\ell} = \sum_{(i, \ell)} |\{j : \pi_{ij} = \ell > \pi_{i+1j}\}| \le \lambda_k. \quad (2)$$

On the other hand, suppose the $k$-th row of $\pi$ has entries $(\ell_1 \ge \cdots \ge \ell_s > 0)$ where $s = \lambda_k$. Assume the entries $\ell_1, \dots, \ell_s$ end in rows $i_1 \ge \cdots \ge i_s$ of $\pi$. Then there is a directed path $\Pi$ from $(k, 1)$ to $(m, n)$ containing all points $(i_s, \ell_s), \dots, (i_1, \ell_1)$. The weight of any such path is at least $\sum_j d_{ij} \ell_j \ge s = \lambda_k$. Combining this with the inequality (2) we obtain (1). $\square$

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**Proof of Theorem 1.** As we are interested only in joint distribution of the last-passage times $(G(m, n), G(m-1, n), \dots, G(1, n))$ we can restrict the source random matrix $W$ to the first $m$ rows and $n$ columns. Consider $w_{ij}$ as geometric with parameter $q_{n-j+1}$. By rotation symmetry it is obvious that the corresponding last-passage times produced from the matrix
$W' = (w_{m-i+1,n-j+1})_{i,j=1}^{m,n}$ have the same distributions. Now, by definition of $g_\lambda$ and using Lemma 2 we obtain that

$$ 
\begin{aligned}
\prod_{\ell=1}^{n} (1 - q_{\ell})^{m} g_{\lambda}(q_{1}, \dots, q_{n}) &= \sum_{\pi \in \text{PP}(n, m), \text{sh}(\pi) = \lambda} \prod_{\ell=1}^{n} (1 - q_{\ell})^{m} q_{\ell}^{c_{\ell}(\pi)} \\
&= \sum_{\pi \in \text{PP}(n, m), \text{sh}(\pi) = \lambda} \text{Prob}(W' = \Phi(\pi))
\end{aligned}
$$

By Lemma 3 if $W' = \Phi(\pi)$ and $\text{sh}(\pi) = \lambda$, we have $G(m, n) = \lambda_1, \dots, G(1, n) = \lambda_m$. Therefore,

$$ \sum_{\pi \in \text{PP}(n, m), \text{sh}(\pi) = \lambda} \text{Prob}(W' = \Phi(\pi)) = \text{Prob}(G(m, n) = \lambda_1, \dots, G(1, n) = \lambda_m) $$

and hence the result is proved.

Next we discuss some applications.

### 3. Last-passage time distributions

#### 3.1. Parameter symmetry

Since the polynomials $g_\lambda$ are symmetric we obtain

**Corollary 4.** *The distribution*

$$ \text{Prob}(G(m, n) = \lambda_1, \dots, G(1, n) = \lambda_m) $$

is invariant under permutations of the parameters $(q_1, \dots, q_n)$.

#### 3.2. Formulas for last-passage distributions

First, using branching formulas for $g_\lambda$, we easily obtain the following distribution formula as well.

**Corollary 5.** *We have*

$$ \text{Prob}(G(m, n) \le \lambda_1, \dots, G(1, n) \le \lambda_m) = \prod_{i=1}^{n} (1 - q_i)^m g_{\lambda}(1, q_1, \dots, q_n). $$

**Proof.** Theorem 1 combined with the branching relation

$$ \sum_{\mu \subseteq \lambda} g_{\mu}(q_1, \dots, q_n) = g_{\lambda}(1, q_1, \dots, q_n) $$

easily imply the given formula.

Using Jacobi–Trudi-type determinantal identities for $g_\lambda$ (see [13]) we get the next formulas.

**Corollary 6.** *The following formulas hold*<fnref n="1" />

$$ 
\begin{aligned}
\text{Prob}(G(m, n) = \lambda_1, \dots, G(1, n) = \lambda_m) &= \prod_{i=1}^{n} (1 - q_i)^m \det \left[ e^{\lambda'_i - i + j} (1^{\lambda'_i - 1}, q_1, \dots, q_n) \right]_{i,j=1}^{\lambda_1} \\
&= \prod_{i=1}^{n} (1 - q_i)^m \det \left[ h^{\lambda'_i - i + j} (1^{i-1}, q_1, \dots, q_n) \right]_{i,j=1}^{m}.
\end{aligned}
$$

**Corollary 7 (Single point distributions via Schur polynomials).** *We have*

$$ \text{Prob}(G(m, n) \le a) = \prod_{i=1}^{n} (1 - q_i)^m s_{(am)}(1^m, q_1, \dots, q_n) $$

<footnote n="1">¹Here $e_n$ is the elementary symmetric polynomials, $h_n$ is the complete homogeneous symmetric polynomials, $1^m = (1, \dots, 1)$ repeated $m$ times, and $\lambda'$ is the conjugate partition of $\lambda$.</footnote>
**Proof.** From Corollary 5 and the first determinantal formula in Corollary 6 which coincides with the Jacobi–Trudi determinant for $s_{(am)}(1^m, q_1, \dots, q_n)$ we have

$$ \mathrm{Prob}(G(m, n) \le a) = \prod_{i=1}^{n} (1 - q_i)^m g_{(am)}(1, q_1, \dots, q_n) = \prod_{i=1}^{n} (1 - q_i)^m s_{(am)}(1^m, q_1, \dots, q_n) $$

as needed.

<figure><img src="image_5.png" /></figure>

**Remark 8.** From this formula, via the Jacobi–Trudi identity one can also obtain Toeplitz as well as Fredholm determinantal expressions using the Borodin–Okounkov formula [2].

**Remark 9.** These distribution formulas were presented in the special i.i.d. case $q_j = q$ in [15].

**Remark 10.** We should also note that there are alternative methods for computing the discussed distributions. There is a well-known connection between last-passage percolation and Schur polynomials, see [4, 6]. Such formulas for the distribution of $G(m, n)$ were first obtained in [4] and used for their asymptotic analysis, see also [6]. In addition, it can be observed that the column distributions $(G(m, n), \dots, G(1, n))$ correspond to distributions of first rows of partitions in appropriately specialized *Schur processes* [9], see also [6]. Here one relies on the Robinson–Schensted–Knuth (RSK) correspondence. The symmetry from Corollary 4 can be seen that way, as well as similar formulas for last-passage time distributions can be derived.

### 4. Generating series identities for $g_\lambda$

By Theorem 1 we can define the probability distribution $P_{m,n}$ on the set of integer partitions $\mathcal{P}_m$ by setting

$$ P_{m,n}(\lambda) := \prod_{i=1}^{n} (1 - q_i)^m g_{\lambda}(q_1, \dots, q_n), \quad \lambda \in \mathcal{P}_m. $$

In particular, since $\sum_{\lambda \in \mathcal{P}_m} P_{m,n}(\lambda) = 1$ we immediately obtain the following identity for dual Grothendieck polynomials (it can also be found in [14, 15]).

**Corollary 11.** The following identity holds

$$ \sum_{\lambda \in \mathcal{P}_m} g_{\lambda}(q_1, \dots, q_n) = \prod_{i=1}^{n} (1 - q_i)^{-m}. $$

Next, observe that we have the following marginal distributions for the parts $\lambda_k$:

$$ P_{m,n}(\lambda_k \le a) = \mathrm{Prob}(G(m-k+1, n) \le a) $$

which give a *shift invariance property*

$$ P_{m,n}(\lambda_k \le a) = P_{m-k+1,n}(\lambda_1 \le a) $$

In particular, the last part $\lambda_m$ has distribution as $\sum_{i=1}^n W_i$ for independent $W_i$ geometrically distributed with parameter $q_i$.

We now present a new more general identity for dual Grothendieck polynomials.

**Theorem 12.** Let $k \in [1, m]$ and $a \in \mathbb{N}$. The following identity holds

$$ \sum_{\lambda \in \mathcal{P}_m, \lambda_k \le a} g_{\lambda}(q_1, \dots, q_n) = \prod_{i=1}^{n} (1 - q_i)^{1-k} s_{(a^{m-k+1})}(1^{m-k+1}, q_1, \dots, q_n). $$

**Proof.** Recall that we have the marginal distributions

$$ P_{m,n}(\lambda_k \le a) = \mathrm{Prob}(G(m-k+1, n) \le a) $$
Using Corollary 7, we have

$$ \mathrm{Prob}(G(m-k+1, n) \le a) = \prod_{i=1}^{n} (1-q_i)^{m-k+1} s_{(a^{m-k+1})} (1^{m-k+1}, q_1, \dots, q_n). $$

On the other hand, by definition of the distribution $P_{m,n}$ above, we get

$$ P_{m,n}(\lambda_k \le a) = \prod_{i=1}^{n} (1-q_i)^m \sum_{\lambda \in \mathcal{P}_m, \lambda_k \le a} g_{\lambda}(q_1, \dots, q_n) $$

Combining the last two identities we obtain the needed. $\square$

**Corollary 13.** For $k=m$ we obtain the following identity

$$ \sum_{\lambda \in \mathcal{P}_m, \lambda_m = a} g_{\lambda}(q_1, \dots, q_n) = \prod_{i=1}^{n} (1-q_i)^{1-m} h_a(q_1, \dots, q_n) $$

**Proof.** For $k=m$ we have

$$ \sum_{\lambda \in \mathcal{P}_m, \lambda_m \le a} g_{\lambda}(q_1, \dots, q_n) = \prod_{i=1}^{n} (1-q_i)^{1-m} h_a(1, q_1, \dots, q_n). $$

Now the following recurrence relation for the polynomials $h$,

$$ h_a(1, q_1, \dots, q_n) - h_{a-1}(1, q_1, \dots, q_n) = h_a(q_1, \dots, q_n) $$

then gives the needed identity. $\square$

**Remark 14.** There is one more connection of dual Grothendieck polynomials with the corner growth model via *positive specializations* of {$g_{\lambda}$}, presented in [17]. The distribution $P_{m,n}$ can also be extended for any positive specialization as we discuss it for a special example in the next section.

**5. Plancherel limit and longest increasing subsequences**

Consider the specialization $q_i = \gamma/n$ for all $i \in [1, n]$, $\gamma > 0$ and let $n \to \infty$. We obtain

$$ \lim_{n \to \infty} P_{m,n}(\lambda) = e^{-m\gamma} \lim_{n \to \infty} g_{\lambda}(\underbrace{\gamma/n, \dots, \gamma/n}_{n \text{ times}}) = e^{-m\gamma} \sum_{n} P_{\mathrm{gpl},m,n}(\lambda) \frac{(m\gamma)^n}{n!}, $$

where $P_{\mathrm{gpl},m,n}(\lambda)$ is a probability distribution on the set of partitions $\lambda \subset (n^m)$, defined below.

To define it, we need a generalization of standard Young tableaux. A plane partition $\pi$ is called a *strict tableau* (ST) if for some $n$, each entry $i \in [n] := \{1, \dots, n\}$ appears in exactly one column of $\pi$. We then say that $[n]$ is a *content* of $\pi$. Let $\mathrm{ST}(\lambda, n)$ be the set of ST of shape $\lambda$ with content $[n]$ and $f_{\lambda}(n) = |\mathrm{ST}(\lambda, n)|$.

**Lemma 15 (see [14]).** We have

$$ P_{\mathrm{gpl},m,n}(\lambda) = \frac{f_{\lambda}(n)}{m^n} $$

is a well-defined probability measure on the set of integer partitions $\lambda \subset (n^m)$.

Let $W_{n,m}$ be the set of words of length $n$ in the alphabet $[m]$. For a word $w = w_1 \cdots w_n \in W_{m,n}$, a *weakly increasing subsequence* is a sequence of the form

$$ w_{i_1} \le \cdots \le w_{i_k}, \quad 1 \le i_1 < \cdots < i_k \le n, $$

where $k$ is its length. Let $L_i(w)$ be the length of the longest weakly increasing subsequence of $w$ using the letters $\{m-i+1, \dots, m\}$. In particular, $L_1(w)$ is just the number of $m$'s in $w$ and $L_m(w)$ is the length of the longest weakly increasing subsequence of $w$.

Consider the uniform probability measure on $W_{n,m}$. Then we have the following analogue of Theorem 1 in this Plancherel limit regime.
**Theorem 16. We have**

$$ \mathrm{Prob}(L_m = \lambda_1, \dots, L_1 = \lambda_m) = P_{\mathrm{gpl},m,n}(\lambda). $$

A combinatorial version of this result (an analogue of Green's theorem for RSK), which can be turned into this statement, is proved in [14]. The distribution of $L_m$ was studied in [12] from which we obtain that for fixed $m$ the limiting distribution of the first row $\lambda_1$ satisfies

$$ \lim_{n \to \infty} P_{\mathrm{gpl},m,n} \left( \frac{\lambda_1 - n/m}{\sqrt{2n/m}} \le t \right) = P_{\mathrm{GUE}_m^0} (\lambda_{\max} \le t), $$

where the r.h.s. is the distribution of the largest eigenvalue in $m \times m$ traceless Gaussian unitary ensemble (GUE). In addition, note that $\lambda_m$ has binomial distribution with parameters $n$ and $1/m$ and hence after proper scaling it converges to normal distribution. Now, what is limiting joint distribution of the properly scaled shape $\lambda$ when $m$ is fixed? This would compare to the results in [5] on limiting distribution of the shape for a random word under the RSK correspondence, which converges to the spectrum of traceless GUE.

**Acknowledgements**

I am grateful to Askar Dzhumadil’daev, Igor Pak, Leonid Petrov, and Pavlo Pylyavskyy for many helpful conversations. I am also grateful to the referee for useful comments suggesting Remark 10.

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