#  $k$-positivity of dual canonical basis elements from 1324- and
2143-avoiding Kazhdan-Lusztig immanants

Sunita Chepuri∗ and  Melissa Sherman-Bennett†

###### Abstract.

In this note, we show that certain dual canonical basis elements of
${\mathbb{C}}[SL_{m}]$ are positive when evaluated on _$k$ -positive
matrices_, matrices whose minors of size $k\times k$ and smaller are positive.
Skandera showed that all dual canonical basis elements of
${\mathbb{C}}[SL_{m}]$ can be written in terms of _Kazhdan-Lusztig immanants_
, which were introduced by Rhoades and Skandera. We focus on the basis
elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by
1324- and 2143-avoiding permutations. This extends previous work of the
authors on Kazhdan-Lusztig immanants and uses similar tools, namely Lewis
Carroll’s identity (also known as the Desnanot-Jacobi identity).

∗University of Michigan, 2074 East Hall, 530 Church Street. Ann Arbor, MI
48109<EMAIL_ADDRESS>
†University of Michigan, 2074 East Hall, 530 Church Street. Ann Arbor, MI
48109<EMAIL_ADDRESS>
##  1\. Introduction

Given a function $f:S_{n}\to{\mathbb{C}}$, the _immanant_ associated to $f$,
$\operatorname{Imm}_{f}X:\text{Mat}_{n\times n}({\mathbb{C}})\to{\mathbb{C}}$,
is the function

$\operatorname{Imm}_{f}X:=\sum_{w\in S_{n}}f(w)~{}x_{1,w(1)}\cdots
x_{n,w(n)},$ (1.1)

where the $x_{i,j}$ are indeterminates. We evaluate $\operatorname{Imm}_{f}X$
on a matrix $M=(m_{i,j})$ by specializing $x_{i,j}$ to $m_{i,j}$ for all
$i,j$.

Immanants are a generalization of the determinant, where
$f(w)=(-1)^{\ell(w)}$, and the permanent, where $f(w)=1$. Positivity
properties of immanants have been studied since the early 1990’s [11, 12, 20,
13]. One of the main results in this area is that when $f$ is an irreducible
character of $S_{n}$, then $\operatorname{Imm}_{f}(X)$ is nonnegative on
_totally nonnegative matrices_ , that is, matrices with all nonnegative minors
[19]. In this note, we will investigate positivity properties of functions
closely related to _Kazhdan–Lusztig immanants_ , introduced by Rhoades and
Skandera [16].

######  Definition 1.1.

Let $v\in S_{n}$. The _Kazhdan-Lusztig immanant_
$\operatorname{Imm}_{v}X:\text{Mat}_{n\times n}({\mathbb{C}})\to{\mathbb{C}}$
is given by

$\operatorname{Imm}_{v}X:=\sum_{w\in
S_{n}}(-1)^{\ell(w)-\ell(v)}P_{w_{0}w,w_{0}v}(1)~{}x_{1,w_{1}}\cdots
x_{n,w_{n}}$ (1.2)

where $P_{x,y}(q)$ is the Kazhdan-Lusztig polynomial associated to $x,y\in
S_{n}$, $w_{0}\in S_{n}$ is the longest permutation, and we write permutations
$w=w_{1}w_{2}\dots w_{n}$ in one-line notation. (For the definition of
$P_{x,y}(q)$ and their basic properties, see e.g. [1].)

Our interest in Kazhdan–Lusztig immanants stems from their connection to the
dual canonical basis of ${\mathbb{C}}[SL_{m}]$. Using work of Du [9], Skandera
[18] showed that the dual canonical basis elements of ${\mathbb{C}}[SL_{m}]$
are exactly Kazhdan–Lusztig immanants evaluated on matrices of indeterminates
with repeated rows and columns.

Let $X=(x_{ij})$ be the $m\times m$ matrix of variables $x_{ij}$ and let
$\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
denote the set of $n$-element multisets of $[m]:=\\{1,\dots,m\\}$. For
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
with $R=\\{r_{1}\leq\cdots\leq r_{n}\\}$ and $C=\\{c_{1}\leq\cdots\leq
c_{n}\\}$, we write $X(R,C)$ to denote the matrix
$(x_{r_{i},c_{j}})_{i,j=1}^{n}$ (see Definition 2.8).

######  Proposition 1.2 ([18, Theorem 2.1]).

The dual canonical basis of ${\mathbb{C}}[SL_{m}]$ consists of the nonzero
elements of the following set:

$\left\\{\operatorname{Imm}_{v}X(R,C):v\in S_{n}\text{ for some
}n\in\mathbb{N}\text{ and
}R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.\right\\}.$

The positivity properties of dual canonical basis elements have been of
interest essentially since their definition, and are closely related to the
study of total positivity. In 1994, Lusztig [14] defined the totally positive
part $G_{>0}$ of any reductive group $G$. He also showed that all elements of
the dual canonical basis of $\mathcal{O}(G)$ are positive on $G_{>0}$. Fomin
and Zelevinsky [10] later proved that for semisimple groups, $G_{>0}$ is
precisely the subset of $G$ where all _generalized minors_ are positive.
Generalized minors are dual canonical basis elements corresponding to the
fundamental weights of $G$ and their images under Weyl group action.

Here, we study signs of dual canonical basis elements on a natural
generalization of $G_{>0}$. Let $S$ be some subset of generalized minors and
$G_{>0}^{S}$ the subset of $G$ where all elements of $S$ are positive. Which
dual canonical basis elements are positive on all elements of $G_{>0}^{S}$? In
this note, we consider the case where $G=SL_{m}$ and $S$ consists of the
generalized minors corresponding to the first $k$ fundamental weights and
their images under the Weyl group action. In this situation, $G_{>0}^{S}$ is
the set of _$k$ -positive matrices_, matrices where all minors of size $k$ and
smaller are positive. Cluster algebra structures, topology, and variation
diminishing properties of these matrices have been previously studied in [2,
4, 8, 7].

We call a matrix functional _$k$ -positive_ if it is positive when evaluated
on all $k$-positive matrices. Our main result is as follows:

######  Theorem 1.3.

Let $v\in S_{n}$ be $1324$-, $2143$-avoiding and suppose that for all $i<j$
with $v_{i}<v_{j}$, we have $j-i\leq k$ or $v_{j}-v_{i}\leq k$. Let
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$.
Then $\operatorname{Imm}_{v}X(R,C)$ is identically zero or it is $k$-positive.

We also characterize precisely when the functions
$\operatorname{Imm}_{v}X(R,C)$ appearing in 1.3 are identically zero (see
Theorem 3.1).

Theorem 1.3 extends the results of [5], in which we showed the function
$\operatorname{Imm}_{v}X([m],[m])$ is $k$-positive under the assumptions of
1.3. Our techniques here are similar to [5]. Note that Theorem 1.3 does not
follow from [5, Theorem 1.4] because for $M$ $k$-positive, $M(R,C)$ is
$k$-nonnegative rather than $k$-positive.

Rephrasing Theorem 1.3 in terms of dual canonical basis elements, we have the
following corollary.

######  Corollary 1.4.

Let $F(X)=\operatorname{Imm}_{v}X(R,C)$ be an element of the dual canonical
basis of ${\mathbb{C}}[SL_{m}]$. Suppose $v$ is $1324$-, $2143$-avoiding and
for all $i<j$ with $v_{i}<v_{j}$, we have $j-i\leq k$ or $v_{j}-v_{i}\leq k$.
Then $F(X)$ is $k$-positive.

The paper is organized as follows. Section 2 gives background on the objects
we will be using to prove Theorem 1.3. It includes several useful lemmas
proven in [5]. Section 3 contains the proof of Theorem 1.3. We conclude with a
few thoughts on future directions in Section 4.

##  2\. Background

In an abuse of notation, we frequently drop curly braces around sets appearing
in subscripts and superscripts.

###  2.1. Background on 1324 and 2143-avoiding Kazhdan-Lusztig immanants

For integers $i\leq j$, let $[i,j]:=\\{i,i+1,\dots,j-1,j\\}$. We abbreviate
$[1,n]$ as $[n]$. For $v\in S_{n}$, we write $v_{i}$ or $v(i)$ for the image
of $i$ under $v$. We use the notation $<$ for both the usual order on $[n]$
and the Bruhat order on $S_{n}$; it is clear from context which is meant. To
discuss non-inversions of a permutation $v$, we’ll write $\langle i,j\rangle$
to avoid confusion with a matrix index or point in the plane. In the notation
$\langle i,j\rangle$, we always assume $i<j$. We use the notation
$\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
for the collection of $n$-element multi-sets of $[m]$. We always list the
elements of a multiset in increasing order.

We are concerned with two notions of positivity, one for matrices and one for
immanants.

######  Definition 2.1.

Let $k\geq 1$. A matrix $M\in\text{Mat}_{n\times n}({\mathbb{C}})$ is _$k$
-positive_ if all minors of size at most $k$ are positive.

An immanant $\operatorname{Imm}_{f}(X):\text{Mat}_{n\times
n}({\mathbb{C}})\to{\mathbb{C}}$ is _$k$ -positive_ if it is positive on all
$k$-positive matrices.

Note that $k$-positive matrices have positive $1\times 1$ minors, i.e.
entries, and so are real matrices.

######  Example 2.2.

The matrix

$M=\begin{bmatrix}22&18&6&3\\\ 8&7&3&2\\\ 2&2&1&2\\\ 1&2&2&6\end{bmatrix}$

is $2$-positive but the upper left $3\times 3$ submatrix has negative
determinant, so is not $3$-positive or $4$-positive (totally positive).

Our results on $k$-positivity of Kazhdan-Lusztig immanants involve pattern
avoidance.

######  Definition 2.3.

Let $v\in S_{n}$, and let $w\in S_{m}$. Suppose $v=v_{1}\cdots v_{n}$ and
$w=w_{1}\cdots w_{m}$ in one-line notation. The pattern $w_{1}\cdots w_{m}$
_occurs_ in $v$ if there exists $1\leq i_{1}<\dots<i_{m}\leq n$ such that
$v_{i_{1}}\cdots v_{i_{m}}$ are in the same relative order as $w_{1}\cdots
w_{m}$. Additionally, $v$ _avoids_ the pattern $w_{1}\cdots w_{m}$ if it does
not occur in $v$.

Certain Kazdhan-Lusztig immanants have a very simple determinantal formula,
which involves the _graph_ of an interval.

######  Definition 2.4.

For $v\in S_{n}$, the _graph_ of $v$, denoted $\Gamma(v)$, refers to its graph
as a function. That is, $\Gamma(v):=\\{(1,v_{1}),\dots,(n,v_{n})\\}$. For
$v,w\in S_{n}$, the graph of the Bruhat interval $[v,w]$ is the subset of
$[n]^{2}$ defined as $\Gamma[v,w]:=\\{(i,u_{i}):u\in[v,w],i=1,\dots,n\\}$.

We think of an element $(i,j)\in\Gamma[v,w]$ as a point in row $i$ and column
$j$ of an $n\times n$ grid, indexed so that row indices increase going down
and column indices increase going right (see 2.6). A _square_ or _square
region_ in $\Gamma[v,w]$ is a subset of $\Gamma[v,w]$ which forms a square
when drawn in the grid.

We will also need the following notion on matrices.

######  Definition 2.5.

Let $P\subset[n]^{2}$ and let $M=(m_{ij})$ be an $n\times n$ matrix. The
_restriction_ of $M$ to $P$, denoted $M|_{P}$, is the matrix with entries

$m^{\prime}_{ij}=\begin{cases}m_{ij}&(i,j)\in P\\\ 0&\text{ else}.\end{cases}$

######  Example 2.6.

Consider $v=2413$ in $S_{4}$. We have
$[v,w_{0}]=\\{2413,4213,3412,2431,4312,4231,3421\\}$, and so $\Gamma[v,w_{0}]$
is as follows.

If $M$ is the matrix from Example 2.2, then

$M|_{\Gamma[v,w_{0}]}=\begin{bmatrix}0&18&6&3\\\ 0&7&3&2\\\ 2&2&1&0\\\
1&2&2&0\end{bmatrix}.$

Note that $v$ avoids patterns 1324 and 2143.

We can now state a simple determinantal formula for certain Kazhdan-Lusztig
elements. This follows from results of [17].

######  Proposition 2.7 ([5, Corollary 3.6]).

Let $v\in S_{n}$ avoid $1324$ and $2143$. Then

$\operatorname{Imm}_{v}(X)=(-1)^{\ell(v)}\det(X|_{\Gamma[v,w_{0}]}).$ (2.1)

Using 2.7, we can similarly obtain a simple determinantal formula for certain
dual canonical basis elements of ${\mathbb{C}}[SL_{m}]$. Recall from 1.2 that
every dual canonical basis element can be expressed as a Kazhdan-Lusztig
immanant evaluated on a matrix of indeterminants with repeated rows and
columns.

######  Definition 2.8.

Let $R=\\{r_{1}\leq r_{2}\leq\dots\leq r_{n}\\}$ and $C=\\{c_{1}\leq
c_{2}\leq\dots\leq c_{n}\\}$ be elements of
$\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
and let $M=(m_{ij})$ be an $m\times m$ matrix. We denote by $M(R,C)$ the
matrix with $(i,j)$-entry equal to $m_{r_{i},c_{j}}$. We call $r_{i}$ the
_label_ of row $i$; similarly, $c_{j}$ is the label of column $j$. We view
$X(R,C)$ as a function from $\text{Mat}_{m\times m}({\mathbb{C}})$ to
$\text{Mat}_{n\times n}({\mathbb{C}})$, which takes $M$ to $M(R,C)$.

Note that our convention is always to list multisets in weakly increasing
order, so the row and column labels of $X(R,C)$ are weakly increasing.

######  Example 2.9.

Let $R=\\{1,1,3\\}$ and $C=\\{2,3,4\\}$. Then

$X(R,C)=\begin{bmatrix}x_{12}&x_{13}&x_{14}\\\ x_{12}&x_{13}&x_{14}\\\
x_{32}&x_{33}&x_{34}\end{bmatrix}.$

If $M$ is the matrix from Example 2.2, then

$M(R,C)=\begin{bmatrix}18&6&3\\\ 18&6&3\\\ 2&1&2\end{bmatrix}.$

We will focus on the dual canonical basis elements
$\operatorname{Imm}_{v}X(R,C)$ where $v$ is 1324- and 2143-avoiding. 2.7
immediately gives a determinantal formula for these immanants.

######  Lemma 2.10.

Let
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
and let $v\in S_{n}$ be $1324$\- and $2143$-avoiding. Then

$\operatorname{Imm}_{v}X(R,C)=(-1)^{\ell(v)}\det X(R,C)|_{\Gamma[v,w_{0}]}.$
(2.2)

We are interested in the sign of $\operatorname{Imm}_{v}X(R,C)$ on
$k$-positive matrices, so long as $\operatorname{Imm}_{v}X(R,C)$ is not
identically zero. Clearly, the function in (2.2) is identically zero when the
matrix $X(R,C)|_{\Gamma[v,w_{0}]}$ has two identical rows or columns. We make
the following definitions to discuss this situation.

######  Definition 2.11.

Let $P\subseteq[n]^{2}$. The _support_ of row $r$ of $P$ is the set of columns
$c\in[n]$ such that $(r,c)\in P$. The support of a column is defined
analogously.

######  Definition 2.12.

Let $P\subseteq[n]^{2}$, and let
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$.
Then $P$ is _$(R,C)$ -admissible_ if no two rows or columns with the same
labels have the same support.

######  Example 2.13.

Let $P=\Gamma[v,w_{0}]$ where $v=2413$, as in Example 2.6. Rows 1 and 2 have
support $\\{2,3,4\\}$ and rows 3 and 4 have support $\\{1,2,3\\}$. Column 1
has support $\\{3,4\\}$, columns 2 and 3 have support $\\{1,2,3,4\\}$, and
column 4 has support $\\{1,2\\}$. This means $P$ is $(R,C)$-admissible if and
only if $r_{1}\neq r_{2},r_{3}\neq r_{4}$, and $c_{2}\neq c_{3}$. For example,
let $A=\\{1,2,2,3\\}$ and $B=\\{1,2,3,3\\}$. Then $P$ is $(A,B)$-admissible
but, since $a_{2}=a_{3}=2$, $P$ is not $(A,A)$-admissible.

For $v$ avoiding 1324 and 2143, $\operatorname{Imm}_{v}X(R,C)$ is identically
zero if $\Gamma[v,w_{0}]$ is not $(R,C)$-admissible. In the subsequent
sections, we will show the converse holds as well (see 3.10).

Finally, we introduce some notation that will be useful in proofs.

For $I\in\binom{[n]}{k}$, define $\delta_{I}:[n]\setminus I\to[n-k]$ as

$\delta_{I}(j):=j-|\\{i\in I:i<j\\}|$

That is, $\delta_{I}$ is the unique order-preserving map from $[n]\setminus I$
to $[n-k]$.

######  Definition 2.14.

For $I,J\in\binom{[n]}{k}$ and $P\subseteq[n]^{2}$, let
$P^{J}_{I}\subseteq[n-k]\times[n-k]$ be $P$ with rows $I$ and columns $J$
deleted. That is, $P^{J}_{I}=\\{(\delta_{I}(a),\delta_{J}(b)):(a,b)\in P\\}$.
The labels of rows and columns are preserved under deletions; to be more
precise, if $R=\\{r_{1}\leq\cdots\leq r_{n}\\}$ is the multiset of row labels
of $P$, the multiset of row labels of $P^{J}_{I}$ is
$\\{r^{\prime}_{1}\leq\cdots\leq r^{\prime}_{n-k}\\}$ where
$r^{\prime}_{j}=r_{\delta_{I}^{-1}(j)}$.

###  2.2. Combinatorics of graphs of upper intervals

We will now take a closer look at the graphs $\Gamma[v,w_{0}]$ that appear in
2.10. We begin by giving an alternate definition for $\Gamma[v,w_{0}]$.

######  Definition 2.15.

Let $v\in S_{n}$ and $(i,j)\in[n]^{2}\setminus\Gamma(v)$. Then $(i,j)$ is
_sandwiched_ by a non-inversion $\langle k,l\rangle$ if $k\leq i\leq l$ and
$v_{k}\leq j\leq v_{l}$. We also say $\langle k,l\rangle$ _sandwiches_
$(i,j)$.

In other words, $(i,j)$ is sandwiched by $\langle k,l\rangle$ if and only if
$(i,j)\in[n]^{2}$ lies inside the rectangle with diagonal corners $(k,v_{k})$
and $(l,v_{l})$.

######  Lemma 2.16 ([5, Lemma 3.4]).

Let $v\in S_{n}$. Then $\Gamma[v,w_{0}]=\Gamma(v)\cup\\{(i,j):(i,j)$ is
sandwiched by a non-inversion of $v\\}$.

Using this alternate characterization, one can translate the assumptions of
1.3 into a condition on $\Gamma[v,w_{0}]$.

######  Lemma 2.17 ([5, Lemma 4.1]).

Let $v\in S_{n}$. The graph $\Gamma[v,w_{0}]$ has a square of size $k+1$ if
and only if for some non-inversion $\langle i,j\rangle$ of $v$, we have
$j-i\geq k$ and $v_{j}-v_{i}\geq k$.

We now introduce some notation and a proposition that we will need to prove
our main result.

######  Definition 2.18.

Let $v\in S_{n}$. Define $\mathbf{B}_{i,v_{i}}$ to be the square region of
$[n]^{2}$ with corners $(i,v_{i}),\ (i,n-i+1),\ (n-v_{i}+1,v_{i})$ and
$(n-v_{i}+1,n-i+1)$. In other words, $\mathbf{B}_{i,v_{i}}$ is the square
region of $[n]^{2}$ with one corner at $(i,v_{i})$ and two corners on the
antidiagonal of $[n]^{2}$. We say $\mathbf{B}_{i,v_{i}}$ is a _bounding box_
of $\Gamma[v,w_{0}]$ if there does not exist some $j$ such that
$\mathbf{B}_{i,v_{i}}\subsetneq\mathbf{B}_{j,v_{j}}$. If
$\mathbf{B}_{i,v_{i}}$ is a bounding box of $\Gamma[v,w_{0}]$, we call
$(i,v_{i})$ a _spanning corner_ of $\Gamma[v,w_{0}]$. (See Figure 1 for an
example.)

Figure 1. An example of $\Gamma[v,w_{0}]$, with
$v=6~{}10~{}4~{}7~{}8~{}9~{}3~{}1~{}2$. The bounding boxes are blue, red,
blue, green, and purple, listed in the order of their northmost row. The
spanning corners of $\Gamma[v,w_{0}]$ are $(1,6)$, $(3,4)$, $(6,9)$, $(8,3)$,
$(9,1)$, and $(10,2)$.

The name “bounding boxes” comes from the following lemma.

######  Lemma 2.19 ([5, Lemma 4.12]).

Let $v\in S_{n}$. Then

$\Gamma[v,w_{0}]\subseteq\bigcup_{(i,v_{i})\in S}\mathbf{B}_{i,v_{i}}.$

We also color the bounding boxes.

######  Definition 2.20.

A bounding box $\mathbf{B}_{i,v_{i}}$ is said to be _red_ if $(i,v_{i})$ is
below the antidiagonal, _green_ if $(i,v_{i})$ is on the antidiagonal, and
_blue_ if $(i,v_{i})$ is above the antidiagonal. If $\mathbf{B}_{i,v_{i}}$ and
$\mathbf{B}_{n-v_{i}+1,n-i+1}$ are both bounding boxes, then
$\mathbf{B}_{i,v_{i}}=\mathbf{B}_{n-v_{i}+1,n-i+1}$ is both red and blue. We
say such a box is _purple_. (See Figure 1 for an example.)

######  Proposition 2.21 ([5, Proposition 4.14]).

Suppose $v\in S_{n}$ avoids $2143$ and $w_{0}v$ is not contained in a maximal
parabolic subgroup of $S_{n}$. Order the bounding boxes of $\Gamma[v,w_{0}]$
by the row of the northwest corner. If $\Gamma[v,w_{0}]$ has more than one
bounding box, then they alternate between blue and red and there are no purple
bounding boxes.

##  3\. Positivity of basis elements

In this section, we prove our main result.

######  Theorem 3.1.

Let
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$,
let $v\in S_{n}$ be $1324$-, $2143$-avoiding and suppose that the largest
square region in $\Gamma[v,w_{0}]$ has size at most $k$. If $\Gamma[v,w_{0}]$
is not $(R,C)$-admissible, then $\operatorname{Imm}_{v}X(R,C)$ is identically
zero. Otherwise, $\operatorname{Imm}_{v}X(R,C)$ is $k$-positive.

Theorem 1.3 easily follows from Theorem 3.1, using Lemma 2.17.

Our proofs rely heavily on Lewis Carroll’s identity.

######  Proposition 3.2 (Lewis Carroll’s Identity).

If $M$ is an $n\times n$ square matrix and $M_{A}^{B}$ is $M$ with the rows
indexed by $A\subset[n]$ and columns indexed by $B\subset[n]$ removed, then

$\det(M)\det(M_{a,a^{\prime}}^{b,b^{\prime}})=\det(M_{a}^{b})\det(M_{a^{\prime}}^{b^{\prime}})-\det(M_{a}^{b^{\prime}})\det(M_{a^{\prime}}^{b}),$

where $1\leq a<a^{\prime}\leq n$ and $1\leq b<b^{\prime}\leq n$.

###  3.1. Young diagram case

We first consider the case where $\Gamma[v,w_{0}]$ is a Young diagram or the
complement of a Young diagram (using English notation). Recall that the
_Durfee square_ of a Young diagram $\lambda$ is the largest square contained
in $\lambda$.

######  Proposition 3.3.

Let $\lambda\subseteq n^{n}$ be a Young diagram with Durfee square of size at
most $k$ and $\mu:=n^{n}/\lambda$. Let $M$ be a $m\times m$ $k$-positive
matrix and
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$.
Then

$(-1)^{|\mu|}\det M(R,C)|_{\lambda}\geq 0$

and equality holds only if $(n,n-1,\dots,1)\nsubseteq\lambda$ or if $\lambda$
is not $(R,C)$-admissible.

###### Proof.

Let $A=M(R,C)|_{\lambda}=\\{a_{ij}\\}$. For $\sigma\in S_{n}$, let
$a_{\sigma}:=a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}$. If
$(n,n-1,\dots,1)\nsubseteq\lambda$ then there is some $1\leq j\leq n$ where
$\lambda_{n-j+1}<j$. Thus boxes in $\lambda$ in the last $j$ rows are in the
southwest most $j\times(j-1)$ rectangle. This means that for every $\sigma$,
$a_{\sigma}$ contains some zero entry, so $\det(A)=0$. It’s clear that if
$\lambda$ is not $(R,C)$-admissable then $\det(A)=0$.

Now we will assume that $(n,n-1,\dots,1)\subseteq\lambda$ and that $\lambda$
is $(R,C)$-admissible. We proceed by induction on $n$ to show that $\det(A)$
has sign $(-1)^{|\mu|}$. The base cases for $n=1,2$ are easy to check.

Let $a=\max\\{i\ |\ \lambda_{i}=n\\}$ and $b=\lambda_{n}=\max\\{j\ |\
\lambda^{\prime}_{j}=n\\}$ where $\lambda^{\prime}$ denotes the transpose of
$\lambda$. In other words, $a$ is the last row in $\lambda$ with $n$ boxes and
$b$ is the last column in $\lambda$ with $n$ boxes. From Lewis Carroll’s
identity, we have that

$\det(A)\det(A_{a,n}^{b,n})=\det(A_{a}^{b})\det(A_{n}^{n})-\det(A_{a}^{n})\det(A_{n}^{b}).$
(3.1)

Let’s see what we know about the signs of these determinants using our
inductive hypothesis. Say $I:=\\{i_{1}<\cdots<i_{k}\\}$ and
$J:=\\{j_{1}<\cdots<j_{k}\\}$, and let $\lambda_{I}^{J}$ denote the Young
diagram obtained from $\lambda$ by removing rows indexed by $I$ and columns
indexed by $J$. Note that

$A_{I}^{J}=M(R,C)_{I}^{J}|_{\lambda_{I}^{J}}=M(R\setminus\\{r_{i_{1}},\dots,r_{i_{k}}\\},C\setminus\\{c_{j_{1}},\dots,c_{j_{k}}\\})|_{\lambda_{I}^{J}}.$

Also, $\lambda_{I}^{J}$ has Durfee square of size at most $k$. So we can use
the inductive hypothesis to compute the signs of all of the determinants in
(3.1) other than $\det(A)$.

Let’s consider which determinants in (3.1) are zero. The shape
$\lambda_{a,n}^{b,n}$ contains the staircase $(n-2,\dots,1)$ and the shapes
$\lambda_{n}^{n},\lambda_{a}^{n}$, and $\lambda_{n}^{b}$ contain the staircase
$(n-1,\dots,1)$. However, $\lambda_{a}^{b}$ may not contain the staircase
$(n-1,\dots,1)$ (e.g. consider $\lambda=(3,3,1)$), so $\det A_{a}^{b}$ may be
zero. Now we need to determine when $\lambda_{I}^{J}$ is
$(R\setminus\\{r_{i_{1}},\dots,r_{i_{k}}\\},C\setminus\\{c_{j_{1}},\dots,c_{j_{k}}\\})$-admissible.
Consider $A_{a,n}^{b,n}$ and pick two row indices $p,q\notin\\{a,n\\}$ with
$p<q$ and $r_{p}=r_{q}$. Because $\lambda$ is $(R,C)$-admissible, rows $p,q$
have different support, so $\lambda_{p}>\lambda_{q}$. Further, because $R$ is
listed in weakly increasing order, $p>a$. We would like to argue that rows
$p^{\prime}:=\delta_{a,n}(p)$ and $q^{\prime}:=\delta_{a,n}(q)$ of
$A_{a,n}^{b,n}$ have distinct support. Since $p>a$, we have
$(\lambda_{a,n}^{b,n})_{p^{\prime}}=\lambda_{p}-1$ and
$(\lambda_{a,n}^{b,n})_{q^{\prime}}=\lambda_{q}-1$, so
$(\lambda_{a,n}^{b,n})_{p^{\prime}}>(\lambda_{a,n}^{b,n})_{q^{\prime}}$. An
analogous argument shows that columns of $A_{a,n}^{b,n}$ with the same index
have different support. Similarly, $A_{a}^{b},A_{a}^{n}$, and $A_{n}^{b}$ are
$(S,D)$-admissible for the appropriate $S,D$. On the other hand, $A_{n}^{n}$
may not be (consider $R=(1,1,2)$, $C=(1,2,3)$, $\lambda=(3,2,1)$, for
example).

Taking all of this together we find that the $\det(A_{a}^{b})\det(A_{n}^{n})$
term in (3.1) may be zero but that $\det(A_{a,n}^{b,n})$ and
$\det(A_{a}^{n})\det(A_{n}^{b})$ are always nonzero. By induction,
$\det(A_{a,n}^{b,n})$ has sign $(-1)^{|\mu|+a+b+1}$ and
$\det(A_{a}^{n})\det(A_{n}^{b})$ has sign $(-1)^{a+b}$. If
$\det(A_{a}^{b})\det(A_{n}^{n})$ is nonzero it has sign $(-1)^{a+b+1}$. Thus,
$\det(A_{a}^{b})\det(A_{n}^{n})-\det(A_{a}^{n})\det(A_{n}^{b})$ always has
sign $(-1)^{a+b+1}$ and $\det(A)$ is always nonzero with sign $(-1)^{|\mu|}$.
∎

######  Corollary 3.4.

Let $\lambda\subseteq n^{n}$ be a Young diagram and let $\mu:=n^{n}/\lambda$.
Suppose $\mu$ has Durfee square of size at most $k$, $M$ is a $k$-positive
$m\times m$ matrix, and
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$.
Then

$(-1)^{|\lambda|}\det M(R,C)|_{\mu}\geq 0$

and equality holds if and only if $(n^{n}/(n-1,n-2,\dots,1,0))\nsubseteq\mu$
(or equivalently, $\lambda\nsubseteq(n-1,n-2,\dots,1,0)$) or if $\mu$ is not
$(R,C)$-admissible.

###### Proof.

Let $\dot{w_{0}}$ denote the matrix with ones on the antidiagonal and zeros
elsewhere. For a multiset $J=\\{j_{1}\leq\cdots\leq j_{n}\\}$, let
$\overline{J}:=\\{\overline{j}_{1}\leq\cdots\leq\overline{j}_{n}\\}$ where
$\overline{j}_{i}:=n+1-j_{n+1-i}$.

Let $M^{\prime}$ be the antidiagonal transpose of $M$; in symbols,
$M^{\prime}=\dot{w_{0}}M^{T}\dot{w_{0}}$. Taking antidiagonal transpose does
not effect the determinant, so $M^{\prime}$ is also $k$-positive.

If we transpose $M(R,C)|_{\mu}$ across the antidiagonal, we obtain the matrix

$N:=M^{\prime}(\overline{C},\overline{R})|_{\nu},$

where $\nu$ is the Young diagram obtained from the skew-shape $\mu$ by
reflecting across the antidiagonal. Applying 3.3, we have that $\det N$ has
sign $|\lambda|$ if $\nu$ is $(\overline{C},\overline{R})$-admissible and is
zero otherwise. It is not hard to check that $\nu$ is
$(\overline{C},\overline{R})$-admissible if and only if $\mu$ is
$(R,C)$-admissible. ∎

We can use 2.7 to rewrite 3.3 and 3.4 in terms of immanants.

######  Corollary 3.5.

Let $v\in S_{n}$ avoid $1324$ and $2143$. Suppose $\Gamma[v,w_{0}]$ is a Young
diagram $\lambda$ with Durfee square of size at most $k$. If $M$ is a
$k$-positive $m\times m$ matrix and
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
such that $\lambda$ is $(R,C)$-admissible, then
$\operatorname{Imm}_{v}M(R,C)>0$.

###### Proof.

Note that $\Gamma(w_{0})\subseteq\Gamma[v,w_{0}]$ implies $\lambda$ contains
the partition $(n,n-1,\dots,1)$. So, by 3.3, we know that $(-1)^{|\mu|}\det
M(R,C)|_{\Gamma[v,w_{0}]}>0$ where $\mu=n^{n}/\lambda$.

Notice that if a box of $\mu$ is in row $r$ and column $c$ then $v(r)<c$ and
$v^{-1}(c)<r$. This means that $(v^{-1}(c),r)$ is an inversion. If $(a,b)$ is
an inversion of $v$ and the box in row $b$ and column $v(a)$ is not in $\mu$,
then $(b,v(a))$ is sandwiched by some non-inversion $\langle a,j\rangle$ for
some $j$. But then $1~{}v(a)~{}v(b)~{}v(j)$ is an occurrence of the pattern
1324, a contradiction. So $(b,v(a))$ is in $\mu$. This means boxes in $\mu$
are in bijection with inversions of $v$ and $(-1)^{\ell(v)}\det
M(R,C)|_{\Gamma[v,w_{0}]}=(-1)^{|\mu|}\det M(R,C)|_{\Gamma[v,w_{0}]}>0$. By
2.7, this means $\operatorname{Imm}_{v}M(R,C)>0$. ∎

######  Corollary 3.6.

Let $v\in S_{n}$ avoid $1324$ and $2143$. Suppose $\Gamma[v,w_{0}]$ is
$\lambda=n^{n}/\mu$ for some partition $\mu$ and the largest square in
$\lambda$ is of size at most $k$. If $M$ is a $k$-positive $m\times m$ matrix
and
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
such that $\lambda$ is $(R,C)$-admissible, then
$\operatorname{Imm}_{v}M(R,C)>0$.

###### Proof.

Note that $\Gamma(w_{0})\subseteq\Gamma[v,w_{0}]$ implies $\lambda$ contains
the partition $(n^{n}/(n-1,n-2,\dots,1,0))$. So, by 3.4, we know that
$(-1)^{|\mu|}\det M(R,C)|_{\Gamma[v,w_{0}]}>0$.

As in the proof of 3.5, there is a bijection between boxes of $\mu$ and
inversions of $v$. So, we know $(-1)^{\ell(v)}\det
M(R,C)|_{\Gamma[v,w_{0}]}=(-1)^{|\mu|}\det M(R,C)|_{\Gamma[v,w_{0}]}>0$. By
2.7, this means $\operatorname{Imm}_{v}M(R,C)>0$. ∎

###  3.2. General Case

The following proposition will allow us to restrict to permutations that are
not elements of a maximal parabolic subgroup of $S_{n}$. To state the lemma we
temporarily denote the longest permutation in $S_{j}$ by $w_{(j)}$.

######  Proposition 3.7 ([5, Corollary 4.9]).

Suppose $v\in S_{n}$ is $1324$-, $2143$-avoiding and $\Gamma[v,w_{0}]$ is
block-antidiagonal. Let $v_{1}\in S_{j}$ and $v_{2}\in S_{n-j}$ be
permutations such that the upper-right antidiagonal block of $\Gamma[v,w_{0}]$
is equal to $\Gamma[v_{1},w_{(j)}]$ and the other antidiagonal block is equal
to $\Gamma[v_{2},w_{(n-j)}]$. Then

$\operatorname{Imm}_{v}M=\operatorname{Imm}_{v_{1}}M([j],[n-j+1,n])\cdot\operatorname{Imm}_{v_{2}}M([j+1,n],[n-j]).$

Figure 2. An example where $\Gamma[v,w_{0}]$ is block-antidiagonal. Here,
$v=74586132$. In the notation of 3.7, $j=3$, $v_{1}=41253$, and $v_{2}=132$.

See Figure 2 for an example illustrating a block-antidiagonal
$\Gamma[v,w_{0}]$ and the notation of Proposition 3.7.

To analyze the determinants appearing in Lewis Carroll’s identity for $\det
X(R,C)|_{\Gamma[v,w_{0}]}$, we will use the following two propositions.

######  Proposition 3.8.

Let $v\in S_{n}$ be $2143$\- and $1324$-avoiding, and choose $i\in[n]$. Let
$x\in S_{n-1}$ be the permutation
$x:\delta_{i}(j)\mapsto\delta_{v_{i}}(v_{j})$ (that is, $x$ is obtained from
$v$ by deleting $v_{i}$ from $v$ in one-line notation and shifting the
remaining numbers appropriately). Then

  1. (1)

$\Gamma[x,w_{0}]=(\Gamma[v,w_{0}]\setminus\\{(p,q):(p,q)\text{ is sandwiched
only by a non-inversion involving }i\\})_{i}^{v_{i}}$;

  2. (2)

If $(i,v_{i})$ is not a spanning corner of $\Gamma[v,w_{0}]$, then
$\Gamma[x,w_{0}]=\Gamma[v,w_{0}]_{i}^{v_{i}}$.

  3. (3)

For all $i$,
$\det(M|_{\Gamma[x,w_{0}]})=\det(M|_{\Gamma[v,w_{0}]_{i}^{v_{i}}}).$

###### Proof.

Statement (1) follows from 2.16. Statements (2) and (3) are Proposition 4.17
from [5]. ∎

######  Proposition 3.9.

Let $v\in S_{n}$ be $1324$\- and $2143$-avoiding such that the last bounding
box of $\Gamma[v,w_{0}]$ is $\mathbf{B}_{n,v_{n}}$, and the second to last box
is $\mathbf{B}_{a,v_{a}}$ for some $a<n$ with $1<v_{a}<v_{n}$. Let
$b=v^{-1}(1)$ and $d=v_{a}$. Suppose
$\det(M(R,C)|_{\Gamma[v,w_{0}]})^{1}_{b}\cdot\det(M(R,C)|_{\Gamma[v,w_{0}]})^{d}_{a}$
is nonzero and has sign $\sigma$. Then

  1. (1)

If
$\det(M(R,C)|_{\Gamma[v,w_{0}]})^{1}_{a}\cdot\det(M(R,C)|_{\Gamma[v,w_{0}]})^{d}_{b}\neq
0$, it has sign $-\sigma$.

  2. (2)

If $\det(M(R,C)|_{\Gamma[v,w_{0}]})^{1,d}_{a,b}\neq 0$, it has sign
$\sigma\cdot(-1)^{\ell(v)}$.

###### Proof.

This follows from the proof of [5, Theorem 4.18]. ∎

We can now determine the sign of $\det X(R,C)|_{\Gamma[v,w_{0}]}$ on
$k$-positive matrices.

######  Theorem 3.10.

Let $v\in S_{n}$ avoid $1324$ and $2143$ and let $k$ be the size of the
largest square in $\Gamma[v,w_{0}]$. Choose
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$.
For $M$ a $k$-positive $m\times m$ matrix,

$(-1)^{\ell(v)}\det M(R,C)|_{\Gamma[v,w_{0}]}\geq 0$

and equality holds if and only if $\Gamma[v,w_{0}]$ is not $(R,C)$-admissible.

###### Proof.

First, if $\Gamma[v,w_{0}]$ is not $(R,C)$-admissible, the determinant in
question is obviously zero. So we assume $\Gamma[v,w_{0}]$ is
$(R,C)$-admissible.

We follow the proof of [5, Theorem 4.18], and proceed by induction on $n$. If
$\Gamma[v,w_{0}]$ is a partition, a complement of a partition, or block-
antidiagonal, we are done by 3.5, 3.6, or 3.7, respectively.

So we may assume that $v$ has at least 2 bounding boxes and that adjacent
bounding boxes have nonempty intersection (where bounding boxes are ordered as
usual by the row of their northeast corner). Because $v$ avoids 1324 and 2143,
the final two bounding boxes of $\Gamma[v,w_{0}]$ are of opposite color by
2.21. Without loss of generality, we assume the final box is red and the
second to last box is blue. Otherwise, we can consider the antidiagonal
transpose of $M(R,C)|_{\Gamma[v,w_{0}]}$. This is equal to
$(\dot{w_{0}}M^{T}\dot{w_{0}})(\overline{C},\overline{R})|_{\Gamma[w_{0}v^{-1}w_{0},w_{0}]}$
(using the notation in the proof of 3.4) and has the same determinant as
$M(R,C)|_{\Gamma[v,w_{0}]}$.

This means the final box is $\mathbf{B}_{n,v_{n}}$, and the second to last box
is $\mathbf{B}_{a,v_{a}}$ for some $a<n$ with $1<v_{a}<v_{n}$. We analyze the
sign of $\det M(R,C)|_{\Gamma[v,w_{0}]}$ using Lewis Carroll’s identity on
rows $a,b:=v^{-1}(1)$ and columns $1,d:=v_{a}$. Note that $a<b$ and $1<d$.

The proof of [5, Theorem 4.18] shows that each of the 5 known determinants in
this Lewis Carroll’s identity is equal to $\det
M(R^{\prime},C^{\prime})|_{\Gamma[v^{\prime},w_{0}]}$ for an appropriate
choice of multisets $R^{\prime},C^{\prime}$ and permutation $v^{\prime}$. We
first show that two of these determinants, forming a single term on the right-
hand side of the identity, are non-zero.

  1. (1)

Consider $(M(R,C)|_{\Gamma[v,w_{0}]})^{1}_{b}$. By 3.8, the determinant of
this matrix is equal to the determinant of
$M(R^{\prime},C^{\prime})|_{\Gamma[y,w_{0}]}$, where $y$ is obtained from $v$
by deleting 1 from $v$ in one-line notation and shifting appropriately,
$R^{\prime}=R\setminus\\{r_{b}\\}$ and $C^{\prime}=C\setminus\\{r_{1}\\}$.

We will check that $\Gamma[y,w_{0}]$ is $(R^{\prime},C^{\prime})$-admissible.
Note that because $(1,b)$ is not a spanning corner of $\Gamma[v,w_{0}]$,
$\Gamma[y,w_{0}]=\Gamma[v,w_{0}]^{1}_{b}$ by 3.8. So we first check that
removing column $1$ and row $b$ from $\Gamma[v,w_{0}]$ does not create any
rows $i,j$ with both the same support and the same labels. By [5, Theorem
4.18, pf. of (2)], rows $b,\dots,n$ of $\Gamma[v,w_{0}]$ all have support
$\\{1,\dots,v_{n}\\}$. Note that removing column $1$ from $\Gamma[v,w_{0}]$
shortens rows $b,\dots,n$ by one and does not effect other rows, so it
suffices to check that rows $b-1,\dots,n-1$ in $\Gamma[y,w_{0}]$ have distinct
labels. Since $\Gamma[v,w_{0}]$ is $(R,C)$-admissible and rows $b,\dots,n$ of
$\Gamma[v,w_{0}]$ have the same support, we must have $r_{b-1}\leq
r_{b}<r_{b+1}<\cdots<r_{n}$. So, letting $r^{\prime}_{i}$ denote the elements
of $R^{\prime}$, indexed in increasing order, we have
$r^{\prime}_{b-1}<r^{\prime}_{b}<\cdots<r^{\prime}_{n-1}$.

We now show there are no columns in $\Gamma[y,w_{0}]$ with both the same
support and same labels. Columns $1,\dots,v_{n}$ of $\Gamma[v,w_{0}]$ have
support containing $[b,n]$, and columns $v_{n}+1,\dots n$ have support
contained in $[1,b-1]$. Removing row $b$ removes one element from the support
of columns $1,\dots,v_{n}$ and does not effect other columns. Any two columns
with the same support in $\Gamma[y,w_{0}]$ correspond to two columns with the
same support in $\Gamma[v,w_{0}]$, and thus have different labels by the
$(R,C)$-admissibility of $\Gamma[v,w_{0}]$.

  2. (2)

Consider $(M(R,C)|_{\Gamma[v,w_{0}]})^{d}_{a}$. By 3.8, the determinant of
this matrix is equal to the determinant of
$M(R^{\prime},C^{\prime})^{d}_{a}|_{\Gamma[z,w_{0}]}$, where $z$ is obtained
from $v$ by deleting $v_{a}$ from $v$ in one-line notation and shifting
appropriately, $R^{\prime}=R\setminus\\{r_{a}\\}$, and
$C^{\prime}=C\setminus\\{c_{d}\\}$. See Figure 3 for an example.

Figure 3. On the left, $\Gamma[v,w_{0}]$ for $v=62785314$. Elements of
$\Gamma[v]$ are marked with crosses. On the right, $\Gamma[z,w_{0}]$ where
$z=5674213$ is the permutation obtained by deleting 2 from the one-line
notation of $v$ and shifting remaining numbers appropriately. Note that
$\Gamma[z,w_{0}]$ is obtained from $\Gamma[v,w_{0}]$ by deleting row 2, column
2, and the shaded region $Q$, consisting of elements sandwiched only by non-
inversions of the form $\langle 2,i\rangle$.

As $(a,v_{a})$ is a spanning corner of $\Gamma[v,w_{0}]$, $\Gamma[z,w_{0}]$ is
obtained from $\Gamma[v,w_{0}]$ by deleting row $a$, column $d$, and the
subset $Q\subset[n]^{2}$ consisting of all elements $(p,q)$ which are
sandwiched only by a non-inversion of the form $\langle a,i\rangle$ (see
Figure 3). Note that if $(p,q)\in Q$, then row $p$ of $\Gamma[v,w_{0}]$ has
support $\\{d,d+1,\dots,d+j\\}$ for some $j$ and column $q$ of
$\Gamma[v,w_{0}]$ has support $\\{a,a+1,\dots,a+\ell\\}$ for some $\ell$.
Notice also that $Q$ consists of some initial chunk of each row and column of
$\Gamma[v,w_{0}]$ it intersects; thus, deleting elements of $Q$ will not
change the largest number in the support of any row or column. Since all
corners of $\Gamma[v,w_{0}]$ are elements of $\Gamma[v]$ and
$(a,d)\in\Gamma[v]$, there are no other corners in row $a$ or column $d$. So
$a$ (resp. $d$) cannot be the largest element in the support of a column
(resp. row). So for row $p$ in $\Gamma[v,w_{0}]_{a}^{d}$, with $\ell$ the
largest element in the support of $p$, $\delta^{-1}_{d}(\ell)$ is the largest
element in the support of row $\delta^{-1}_{a}(p)$ in $\Gamma[v,w_{0}]$. An
analogous statement holds for column $q$ in $\Gamma[v,w_{0}]_{a}^{d}$.

Consider rows $p<p^{\prime}$ of $\Gamma[v,w_{0}]$ with $r_{p}=r_{p}^{\prime}$
and $p,p^{\prime}\neq a$. Because $R$ is listed in weakly increasing order,
$r_{p}=r_{p+1}=\cdots=r_{p^{\prime}}$. By assumption, the support of these
rows in $\Gamma[v,w_{0}]$ must be different. Suppose rows $s=\delta_{a}(p)$,
$s^{\prime}=\delta_{a}(p^{\prime})$ have the same support in
$\Gamma[z,w_{0}]$; say $\ell$ is the largest number in their support. The
reasoning in the above paragraph implies that $\delta^{-1}_{d}(\ell)$ is the
largest number in the support of rows $p,p^{\prime}$ in $\Gamma[v,w_{0}]$, and
thus also in rows $p,p+1,\dots,p^{\prime}-1,p^{\prime}$. So the smallest
number in the support of rows $p,p+1,\dots,p^{\prime}$ must be different. On
the other hand, after deleting column $d$ and the elements of $Q$, the
supports should be the same. These deletions remove the first element of a row
only if that first element is in column $d$. Putting these together, we must
have $p=a-1$, $p^{\prime}=a+1$, and row $a+1$ has support starting at $d$;
otherwise we obtain rows of $\Gamma[v,w_{0}]$ with the same label and same
support. But now row $a$ is among rows $p,p+1,p^{\prime}$, and rows $a$ and
$p^{\prime}=a+1$ have support starting at $d$, a contradiction.

An identical argument with columns in place of rows shows that no two columns
of $\Gamma[z,w_{0}]$ have the same support and the same label. So
$\Gamma[z,w_{0}]$ is $(R^{\prime},C^{\prime})$-admissible.

So by the inductive hypothesis, one term on the right-hand side of the
identity is nonzero. Let $\sigma$ denote the sign of this term. By 3.9, the
other term on the right-hand side has sign $-\sigma$ if it is nonzero. In
either case, the right-hand side has sign $\sigma$, and in particular is
nonzero. Thus, both determinants on the left-hand side are non-zero. By 3.9,
the determinant $\det(M(R,C)|_{\Gamma[v,w_{0}]})^{1,d}_{a,b}$ has sign
$\sigma\cdot(-1)^{\ell(v)}$, so dividing through by that determinant shows
that $\det M(R,C)|_{\Gamma[v,w_{0}]}$ has sign $\ell(v)$.

∎

Taking this theorem with 2.10, we can now prove 3.1.

###### Proof of 3.1.

By 2.10,

$\operatorname{Imm}_{v}M(R,C)=(-1)^{\ell(v)}\det M(R,C)|_{\Gamma[v,w_{0}]}.$

Let $k^{\prime}\leq k$ be the size of the largest square in $\Gamma[v,w_{0}]$.
By 3.10, for $M$ $k^{\prime}$-positive, the right hand side of this expression
is positive. Any $k$-positive matrix is also $k^{\prime}$-positive, so we are
done. ∎

##  4\. Future Directions

The results in [5] and this paper were inspired by the following conjecture of
Pylyavskyy.

######  Conjecture 4.1 ( [15]).

Let $0<k<n$ be an integer and let $v\in S_{n}$ avoid the pattern
$12\cdots(k+1)$. Then $\operatorname{Imm}_{v}X$ is $k$-positive.

This conjecture remains open. The relation between pattern avoidance and
$k$-positivity of immanants is an interesting direction of further inquiry.

The results of this paper showcase an interesting phenomenon: the behavior of
the dual canonical basis element $\operatorname{Imm}_{v}X(R,C)$ on
$k$-positive matrices is the same as the behavior of the usual Kazhdan-Lusztig
immanant $\operatorname{Imm}_{v}X$. Based on this, we make the following
conjecture.

######  Conjecture 4.2.

Suppose $\operatorname{Imm}_{v}X$ is $k$-positive. Then as long as
$\operatorname{Imm}_{v}X(R,C)$ is not identically zero,
$\operatorname{Imm}_{v}X(R,C)$ is $k$-positive.

We also make a related conjecture based on the same phenomenon, which is
something of an intermediate conjecture; it would imply Conjecture 4.1 and
would be implied by Conjectures 4.1 and 4.2 together.

######  Conjecture 4.3.

Let $0<k<n\leq m$ be integers and let $v\in S_{n}$ avoid the pattern
$12\cdots(k+1)$. Let
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$.
If $\operatorname{Imm}_{v}X(R,C)$ is not identically zero, then
$\operatorname{Imm}_{v}X(R,C)$ is $k$-positive.

The compact determinantal formulas we give for certain dual canonical basis
elements may be useful to understand the relationship between the dual
canonical basis of ${\mathbb{C}}[SL_{m}]$ and its cluster algebra structure.
Technically, the cluster algebra in question is the coordinate ring of
$G^{w_{0},w_{0}}$, the open double Bruhat cell in $SL_{m}$;
${\mathbb{C}}[G^{w_{0},w_{0}}]$ differs from ${\mathbb{C}}[SL_{m}]$ by
localization at certain principal minors. The cluster monomials of
${\mathbb{C}}[G^{w_{0},w_{0}}]$ are expected to be dual canonical basis
elements. One natural question is: do the cluster monomials include the
functions $\operatorname{Imm}_{v}X(R,C)$, where $v$ avoids 2143 and 1324? If
so, can the $k$-positivity of these immanants be explained from a cluster
algebraic viewpoint?

Work related to these questions appeared in the manuscript [6]; the connection
to Kazhdan-Lusztig immanants is explained in [3, Section 3.3]. The results of
[6] show that $\operatorname{Imm}_{v}X(R,C)$ is a cluster variable for $v$
avoiding 123, 2143, 1432, and 3214. The immanants occurring in [6] have a
determinantal form given by 2.10; they further conjecture that all cluster
variables of ${\mathbb{C}}[G^{w_{0},w_{0}}]$ can be written as $\pm\det
X(R,C)|_{P}$ for some $P\subset[n^{2}]$. Conjecturally, the Kazhdan–Lusztig
immanants that can be written as $\pm\det X(R,C)|_{P}$ are the exactly
$\operatorname{Imm}_{v}X(R,C)$ where $v$ is 2143 and 1324 avoiding. This leads
to the following conjecture.

######  Conjecture 4.4.

Fix $m$ and let $G^{w_{0},w_{0}}$ denote the big open double Bruhat cell in
$SL_{m}$.

  1. (1)

All cluster variables of ${\mathbb{C}}[G^{w_{0},w_{0}}]$ are of the form
$\operatorname{Imm}_{v}X(R,C)$ for some $v$ avoiding $2143$ and $1324$.

  2. (2)

For $v\in S_{n}$ avoiding $2143$ and $1342$ and
$R,C\in\left.\mathchoice{\left(\kern-4.79996pt\binom{[m]}{n}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[m]}}{\smash{n}}\kern-3.00003pt\right)}\right.$
with $\Gamma[v,w_{0}]$ $(R,C)$-admissible, $\operatorname{Imm}_{v}X(R,C)$ is a
cluster variable in ${\mathbb{C}}[G^{w_{0},w_{0}}]$ if it is irreducible and a
cluster monomial otherwise.

##  5\. Acknowledgements

We would like to thank Pavlo Pylyavskyy for suggesting this topic to us. This
material is based upon work supported by the National Science Foundation under
Grant No. DMS-1439786 while the first author was in residence at the Institute
for Computational and Experimental Research in Mathematics in Providence, RI,
during the Spring 2021 semester. The second author was supported by an NSF
Graduate Research Fellowship DGE-1752814.

## References

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  * [6] M. Chmutov, P. Jiradilok, and J. Stevens, Double rim hook cluster algebras, REU Report, (2015).
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  * [15] P. Pylyavskyy, personal communication, November 2018.
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††thanks: Current Adress: Simbeyond B.V., Het Eeuwsel 57, AS Eindhoven 5612,
Netherlands

# Surface termination dependence of electronic and optical properties in
Ti2CO2 MXene monolayers

Zafer Kandemir  Department of Mechanical Engineering, Faculty of Engineering,
Eskisehir Technical University, 26555, Eskisehir, Turkey  Engin Torun  Fulvio
Paleari  Istituto di Struttura della Materia and Division of Ultrafast
Processes in Materials (FLASHit) of the National Research Council, via Salaria
Km 29.3, I-00016 Monterotondo Stazione, Italy.  Celal Yelgel  Department of
Electricity and Energy, Recep Tayyip Erdogan University, 53100, Rize, Turkey
Cem Sevik  Department of Mechanical Engineering, Faculty of Engineering,
Eskisehir Technical University, 26555, Eskisehir, Turkey
<EMAIL_ADDRESS>
###### Abstract

Two-dimensional (2D) MXenes are a rapid growing family of 2D materials with
rich physical and chemical properties where their surface termination plays an
essential role. Among the various 2D MXenes, functionalization of the TinCn-1
phase with oxygen (O) atoms makes them attractive for optoelectronic
applications due to their optical gap residing in the infrared or visible
region. In this manuscript, we theoretically investigate the electronic and
optical properties of four different O-atom-functionalized TinCn-1 MXene
monolayers using state-of-the-art, first-principles techniques. In particular,
we calculate the quasiparticle corrections on top of density functional theory
(DFT) at the GW level and the exciton-dominated optical spectra by solving the
Bethe-Salpeter equation (BSE) also at finite momentum. We find that all but
one of the monolayer models are indirect band gap semiconductors where
quasiparticle corrections are very important ($\sim 1$ eV). The optical
spectra are instead dominated by direct and indirect excitons with large
binding energies (between $0.5$ and $1$ eV). Most direct excitons lie above
$1.5$ eV, while the indirect ones are below: therefore, we conclude that
TinCn-1 should display strong absorption in the visible region, but phonon-
assisted emission in the infrared. Our work thus reveals the potential usage
of surface terminations to tune the optical and electronic properties of
TinCn-1 MXene monolayers, while emphasizing the pivotal role of many-body
effects beyond DFT to obtain accurate prediction for these systems.

##  I Introduction

The family of 2D transition metal carbide, nitride, and carbonitride materials
– the so-called “MXenes” – possessing chemical formula of Mn+1XnTy ($n$=1, 2
or 3), where “M” is an early transition metal such as Sc, Ti, Zr, Hf, V, Nb,
Ta, Cr, Mo or W, “X” is either N or C, and “Ty” stands for the surface
terminations such as O, OH, F or S has been the object of great interest since
its first introduction by Gogotsi et al.[1] These layered materials are mostly
chemically synthesized through selective acid etching of A elements from MAX
phases, [2, 3, 4, 5] where ”A” is a group IIIA to VIA element. In addition,
chemical transformations and bottom-up construction techniques[6] such as
chemical vapor deposition have been also demonstrated for the successful
synthesis of some MXene crystals. To date, many kinds of MXene crystals have
been experimentally realized[6, 7, 8, 9, 10] and methods to control formation
on surface termination have been demonstrated as well. [11, 12] Indeed, large-
scale single-layer Ti2CO2 crystals have not been reported along with proper
characterization of a physical property such as optical. However, due to the
enormous research effort and current developments on techniques for MXene
Delamination into Single-Layer Flakes it is close to coming true[13]. On the
other hand, single-layer crystals with random functional groups such as OH, F,
O, Cl are available in the literature[14].

Numerous different MXene layered systems arise by combining their chemical
versatility and thier wide range of surface functionalizations, as
demonstrated in research studies revealing the enormous potential of MXenes in
various applications[8, 9, 15] such as power generation and storage, [6, 10,
16, 17, 18] gas, piezoresistive and strain sensors, [19, 20, 21, 22, 23]
chemical catalysis, [7] water purification, [24, 25, 26] plasmonics, [27, 28,
29] transparent conductors[30] and electromagnetic interference shielding.[31]
Recent studies have also suggested the occurrence of superconductivity and of
magnetic properties which might be the subject of future qubit and skyrmion-
based investigations. [32, 11, 33, 34]

Among the experimentally available MXenes, TinCn-1Tn is the most widely
investigated one due to its largest superficial area per weight and being one
of the thinnest MXene phases. [35] First-principles calculations have shown
that the pristine TinCn-1 are metallic. [3] However, after functionalization
with O, 2D Ti2C becomes semiconducting with a considerable band gap energy,
[36] this also holds for the Zr2CO2 and Hf2CO2 phases, as well.[37]

The notable influence of the surface termination on the physical properties,
e.g., electronic, mechanical, ionic diffusion, and ionic absorption have
already been investigated for O-terminated Ti2C monolayers.[38, 39, 40, 41]
The effect of surface termination on the optical properties of these
materials, on the other hand, has not been systematically investigated and the
literature is rather sparse. For instance, the optical gap and the binding
energy of the corresponding first bright direct exciton have been reported
using GW and BSE formalism for the most chemically stable O-terminated Ti2C
monolayer as 1.26 eV and 0.56 eV, respectively. [42] The absorbed photon flux
has been calculated as 1.76 mAcm-2 which is comparable with the 1 nm thick
layers of Si (0.1 mAcm-2), GaAs (0.3 mAcm-2), and P3HT polymer (0.2 mAcm-2).
[42, 43] This consequently emphasizes the potential of using these monolayers
in photodetection and photovoltaic applications. On the other hand, most of
the semiconductor MXene structures have been determined as indirect band gap
semiconductors, [36, 37] which means that it is important to include indirect
excitons in the computational studies to predict the optical properties and
application areas of the different MXene structures. Taking into consideration
all these facts, in this manuscript we present a thorough analysis based on
first-principles calculations on the ground and excited state properties of
the four different O terminated models of Ti2CO2 monolayers. We first present
the electronic structures of the monolayer models and show that all of them
but one are indirect band gap semiconductors at both DFT and GW levels.
Subsequently, we investigate the optical properties of these monolayers
including direct and indirect excitons by solving BSE using quasi particle
(QP) energies and DFT wave functions. We observe that the bound indirect
excitons are the lowest lying ones in the absorption spectra of indirect
monolayer models although their binding energies are in average lower than the
direct ones.

This manuscript is organized as follows: we first give the computational
details in Sec. II. Then in Sec. III we present and discuss the results
corresponding to the DFT, GW, and BSE calculations. Finally, we summarize our
main findings in Sec. IV.

##  II Computational methods

The DFT ground state calculations were performed with Quantum ESPRESSO[44, 45]
using the local density approximation[46] (LDA) and norm-conserving
pseudopotentials. [47] The energy cutoff for the plane wave basis was set to
$90$ Ry and a $\Gamma$-centered 18$\times$18$\times$1 k-point mesh was used,
which guarantees a total energy convergence of 10-10 eV during the self-
consistent steps. The geometries were fully optimized until the Hellmann-
Feynman forces on each atom were less than $0.02$ eV/Å. A vacuum separation of
20 Å in the out-of-plane direction was ensured to eliminate spurious
interactions between monolayers with their periodically repeated images. Spin-
orbit coupling is not taken into account in the simulations presented in this
manuscript as our tests revealed its effects being negligible for the systems
under investigation.

The many-body perturbation theory (MBPT) calculations, performed on top of the
DFT results, were conducted with the YAMBO code.[48, 49] The G0W0[50]
corrections to the Kohn-Sham eigenvalues were computed with a plasmon-pole
approximation for the dynamical electronic screening. The direct and indirect
band gaps were converged with a 48 $\times$ 48 $\times$ 1 $k$-grid mesh, 217 k
points in the irreducible Brillouin zone (BZ), summing over $400$ states for
both the screening function and the Green’s function. The corrections were
computed for the top 3 valence bands and the bottom 3 conduction bands. The
BSE[bse-ref2] was then solved with RPA static screening, which was summed over
$400$ bands and in the Tamm-Dancoff approximation on top of the GW results.
The direct and indirect (i.e. finite-momentum) exciton energies and their wave
functions were obtained for the first 5500 excitonic states by using the
iterative scheme enabled by the SLEPC library.[51] The Coulomb cutoff (CC)
technique was used along the out-of-plane direction to eliminate the
interactions to all the undesired images of the systems in both G0W0 and BSE
steps.[52] Convergence tests for the parameters used in MBPT calculations are
provided in the supplementary material[smaterials]. 111The input parameters
were individually converged by slowly increasing them until differences in
band gaps and exciton energies between two subsequent runs were below 0.1 eV

Figure 1: (Color online) Top and side views of the optimized crystal
structures of the O-terminated Ti2CO2 monolayers. A $3\times 3$ supercell is
shown for clarity. The blue, red and brown spheres represent titanium, oxygen
and carbon atoms, respectively. (a) O-hTi; O atoms in the hollow site between
Ti atoms and on top of the Ti atom in the opposite layer. (b) O-hC; O atoms
again in the hollow site, but this time both of them are on top of the C atom.
(c) O-Ti; both O atoms on Ti atoms. (d) O-hTiC; O atoms in the hollow sites,
but with one O atom above the C atom and the other O atom above the opposite
Ti atom.

##  III Results and Discussion

Crystal structure of the four possible O-terminated hexagonal Ti2C monolayer
models are shown in Fig. 1(a-d). The corresponding binding energies of the O
atom are predicted via the following equation:

$E_{b}=\dfrac{1}{N}\big{[}E_{\mathrm{Ti}_{2}\mathrm{CO}_{2}}-E_{\mathrm{Ti}_{2}\mathrm{C}}-2E_{\mathrm{O}}]$
(1)

where $E_{\mathrm{Ti}_{2}\mathrm{CO}_{2}}$, $E_{\mathrm{Ti}_{2}\mathrm{C}}$,
and $E_{\mathrm{O}}$ are the total energies of Ti2CO2, Ti2C, and the isolated
O atom, respectively ($N$ being the total number of atoms in the unit cell).

The calculated binding energies ($E_{b}$) along with lattice constants ($a$)
and thicknesses ($d$) of all the investigated Ti2CO2 monolayers are reported
in Table 1. The results are in good agreement with recent works based on DFT
calculations with different functionals.[42, 36, 24, 54] Here, more negative
binding energy indicates the more favorable exothermic binding of O atoms.
Therefore, the most and the least chemically stable structures are predicted
as O-hTi and O-Ti, respectively. We should note that these results are only to
compare the chemical stability of these structures in their pristine form and
any one of them could be stabilized by temperature, pressure, growth
conditions and substrate effects.

Table 1: Calculated parameters of the O-terminated Ti2CO2 monolayers: Lattice constant ($a$), monolayer thickness ($d$), binding energy of the O atom ($E_{b}$), band gap energies from the LDA ($E_{gap}^{LDA}$) and GW calculation ($E_{gap}^{GW}$), location of the valence band maximum (VBM), conduction band minimum (CBM) in the BZ and type of the band gap. Note that the minimum direct band gap of the indirect monolayers are reported in the paranthesis. System |  $a$ (Å) |  $d$ (Å) |  $E_{b}$ (eV/atom) |  $E_{gap}^{LDA}$ (eV) |  $E_{gap}^{GW}$ (eV) | VBM | CBM | Type
---|---|---|---|---|---|---|---|---
O-hTi | 2.98 | 4.35 | -4.97 | 0.28(0.61) | 1.29(1.86) | $\Gamma$ | $M$ | Indirect
O-hC | 2.90 | 4.75 | -4.59 | 0.36(0.61) | 1.19(1.42) | $K-\Gamma$ | $\Gamma-M$ | Indirect
O-Ti | 3.28 | 5.43 | -3.83 | 0.68 | 1.92 | $M$ | $M$ | Direct
O-hTiC | 2.95 | 4.51 | -4.79 | 0.74(1.30) | 1.74(2.46) | $\Gamma$ | $M$ | Indirect
Figure 2: (Color online) Band structures of Ti2CO2 monolayers: (a) O-hTi, (b)
O-hC, (c) O-Ti and (d) O-hTiC. The light blue dashed and red dotted lines
represent LDA and G0W0 band structures, respectively. The LDA band structures
are shifted to the GW band gap to compare the band dispersions. The black
dashed line indicates the Fermi level which is shifted to 0 eV.

###  III.1 Electronic structure and quasiparticles

The pristine bulk Ti2C (non-terminated) is metallic with a high density of
states at the Fermi level.[55] However, it turns into a semiconductor when
terminated with O atoms. In order to address the DFT-LDA band gap
underestimation which leads to discrepancies between calculated and
experimental spectra, [42, 36] we performed GW calculations to access the QP
spectral properties. Fig.2 demonstrates the LDA and GW-corrected band
structures of the Ti2CO2 monolayers. It is important to note that LDA gaps are
shifted to the GW ones to compare the dispersion of the bands. It is observed
that band dispersions are very similar at both LDA and GW levels for the
valence bands but slightly different for the conduction bands particularly
along $K-\Gamma$ direction. At both LDA and GW level, O-hTi, O-hC, and O-hTiC
monolayers are found to be indirect but O-Ti a direct band gap semiconductor
with a GW (LDA) band gap of 1.29 (0.28), 1.19 (0.36), 1.74 (0.74) eV, and 1.92
(0.68), respectively as reported in Tab. 1 together with the other properties
of the monolayers. As can be seen that the band gap values are enormously
increased by the self-energy corrections and brings them much closer to the
low-energy side of the visible spectrum. In particular, the indirect QP band
gap of O-hTi is now 1.29 eV, more than four times its DFT value, while the
direct O-Ti gap is now 1.92 eV, increasing by almost three times. Our results
are in good agreement with the available results for O-hTi structure, 1.32[42]
and 1.15[36] eV. The indirect band gap of O-hTi is 0.28 eV at the LDA level
also agrees well with the reported ones which are in the range of 0.24 and
0.32 eV. [24, 56, 36, 42, 57, 58]

Figure 3: (Color online) Total and partial densities of states (DOS) of Ti2CO2
monolayers: (a) O-hTi, (b) O-hC, (c) O-Ti and (d) O-hTiC. The total DOS, and
partial DOSes of Ti-3d, C-2p and O-2p orbitals are shown in black, red short-
dashed, green long-dashed and blue dotted-dashed lines, respectively. Fermi
energy is shifted to 0 ev.

Although having exactly the same atomic composition, the GW correction to the
LDA band gap varies and resulting QP gaps do not follow the same energy
ordering as in the DFT case due to the different screening environment for the
charge carrier interaction in each monolayer. The largest correction to the
band gap, 1.24 eV, is calculated for the O-Ti monolayer, which might be
expected as (i) the higher localisation of the isolated O orbitals leads to
stronger corrections, and (ii) O-Ti is the thinnest one among the considered
systems, therefore the electrons here are even more weakly screened compared
to the other cases, again leading to larger band gap openings.

Figure 4: (Color online) The imaginary part of the dielectric functions -
proportional to the absorption spectrum - of the O-terminated Ti2CO2 monolayer
models: (a) O-hTi, (b) O-hC, (c) O-Ti and (d) O-hTiC. The blue solid and blue
dashed lines represent the spectrum computed with GW+BSE and GW+IPA methods,
respectively. The solar flux of terrestrial (AM1.5g) spectra and visible-light
region is shown in the background[59]. The energy positions of the lowest-
lying finite-momentum excitons are shown with vertical dashed lines for
comparison. The exciton states are labelled as in the main text. Insets:
location in the BZ of the most relevant electron-hole contributions to the
labelled excitons.

Partial DOS analysis (Fig.3) reveals that the 3d orbitals of Ti and 2p
orbitals of C and O atoms partially contribute to the valence and conduction
bands of Ti2CO2 monolayers. Rather large DOS around Fermi level and conduction
band minimum of the O-hC, O-Ti and O-hC monolayers can be noticed in the
figure. This ultimately leads to high joint DOS which is an indication of
having strong light absorption and emission properties of these monolayers. On
the other hand, the very dispersive valence bands of O-hTi monolayer leads to
drastically reduced DOS around Fermi level. This suppresses the joint DOS and
hence the optical response of the monolayer.

In view of these striking results, it clearly becomes necessary to investigate
the effects of electron-hole interaction in order to accurately determine the
absorption and emission energy ranges of O-terminated layered Ti2CO2 systems.

###  III.2 Optical properties and excitons

The imaginary part, $\epsilon_{2}(\omega)$, of the frequency dependent
dielectric function
$(\varepsilon(\omega)=\varepsilon_{1}(\omega)+i\varepsilon_{2}(\omega))$
proportional to the optical absorption spectrum which is defined in the
independent-particle approximation (IPA) as [60]

$\varepsilon_{2}(\omega)=\dfrac{8\pi^{2}e^{2}}{V}\sum_{\kappa}|d_{\kappa}|^{2}\delta(\omega-\Delta\epsilon_{\kappa})$
(2)

Here, $d_{\kappa}$ is the dipole matrix element and $\Delta\epsilon_{\kappa}$
is the transition energy of the electrons that absorb the incoming
electromagnetic field with frequency $\omega$. When the transition is allowed
by the symmetry, a peak appears in the absorption spectrum at the transition
energy with an intensity proportional to the transition probability.

This approach might provide reasonable absorption spectra for the materials
where the Coulomb interaction is highly screened and hence electron-hole
interaction has negligible effect on the optical response of the material.
However, for the very thin materials, such as O-terminated Ti2CO2 monolayers,
the screening of the Coulomb interaction is drastically reduced due to the
absence of screening in vacuum which in turn enhances the electron-hole
interaction. Therefore, it is necessary to plug in the electron-hole
interaction into $\epsilon_{2}(\omega)$ (Eqn. 2) via MBPT as

$\varepsilon_{2}(\omega)=\dfrac{8\pi^{2}e^{2}}{V}\sum_{\lambda}|\sum_{\kappa}\bar{A}^{\kappa}_{\lambda}d_{\kappa}|^{2}\delta(\omega-
E_{\lambda})$ (3)

Here excitations are summed over excitons, $\lambda$, which are composed of
linear combination of single-particle transitions, $\kappa$, with weights,
$\bar{A}^{\kappa}_{\lambda}$, and energy, $E_{\lambda}$. All these excitonic
quantities can be computed by solving the BSE. It is important to note that
only “vertical” or “direct” ($q=k_{v}-k_{c}=0$) excitons are relevant for the
optical absorption. On the other hand, finite-momentum excitons are
particularly decisive for the emission profiles of indirect semiconductor
systems such as some of the Ti2CO2 monolayer models studied in this
manuscript. Therefore, in these systems we solve the BSE also at the finite
$q$ corresponding to their indirect band gaps (as reported in Tab. 1) in order
to gain insight on their optical emission features.

Table 2: Energies of direct (D) and indirect (ID) excitons of the O-terminated Ti2CO2 monolayers with their respective binding energies in parentheses. All values are in eV. System | $\rm D_{1}$ | $\rm D_{2}$ | ${\rm ID}_{1}$ | ${\rm ID}_{2}$
---|---|---|---|---
O-hTi | 1.39 (0.51) | 2.45 (0.76) | 0.80 (0.49) | –
O-hC | 0.79 (0.98) | 1.46 (0.77) | 0.63 (0.56) | 0.67 (0.52)
O-Ti | 1.20 (0.84) | – | – | –
O-hTiC | 2.02 (0.73) | – | 1.15 (0.59) | –

The imaginary part of the dielectric function of the Ti2CO2 monolayer models
are shown in Fig. 4 which are calculated using the QP energies and LDA wave
functions. Spectra of the monolayers on top of the LDA eigenvalues are
provided in the supplementary material of the manuscript for comparison. The
solid and dashed blue lines in each figure corresponds to the spectrum with
(GW+BSE) and without (GW+IPA) excitons, respectively. Note that plotted
spectra correspond to “vertical” or “direct” ($q=k_{v}-k_{c}=0$) excitons, we
indicate the bound “indirect” excitons in the figures as dashed vertical lines
where relevant. It is known that the exciton is a collective excitation,
meaning that all electronic transitions in principle contribute to the
excitonic peak, albeit with a certain weight. We provide the transitions that
have the largest contribution to the specific excitonic peak in the BZ as
subfigure for each monolayer. The binding energy of the excitons are reported
in Tab. 2 and calculated as the energy difference between the exciton energy
and the QP single particle transition energy with greatest weight and same
momentum.

####  III.2.1 O-hTi monolayer

O-hTi monolayer is an indirect band gap semiconductor with a band gap of 1.29
eV at the GW level (Tab. 1), where the VBM and CBM are at $\Gamma$ and $M$
points in the BZ, respectively (Fig. 2(a)). Two prominent excitonic peaks D1
and D2 at 1.39 and 2.45 eV can be identified in the optical spectrum (Fig.
4(a)) and among those peaks, D1 has a binding energy of 0.51 eV with
corresponding transitons around $\Gamma$ point in the BZ as shown in the
subfigure. The transitions which composed of the D2 exciton are, on the other
hand, reside along the $\Gamma-M$ direction in the BZ as shown in the
subfigure. Change in the wave function characteristics of the contributed
orbitals manifest itself in the binding energy of the D2 exciton which is
calculated as 0.76 eV and rather larger than that of D1. In addition to these
vertical excitons, we also indicate the energy, 0.80 eV, of the lowest energy
indirect exciton (ID1) in the figure as a grey vertical dashed line. It is
found that the ID1 exciton is the lowest energy exciton of the O-hTi monolayer
with a binding energy of 0.49 eV for which the largest weight transitions
reside between $\Gamma$ and $M$ points in the BZ. Upon comparison with the
solar flux of terrestrial spectra, it is expected that O-hTi monolayer has
strong absorption in the near infrared and visible (blue and green color) but
indirect emission in the deep infrared region.

####  III.2.2 O-hC monolayer

Similar to O-hTi, O-hC monolayer is also an indirect band gap semiconductor
with a band gap of 1.42 eV at the GW level (Tab. 1) where the VBM and CBM
reside in between K-$\Gamma$ and $\Gamma$-$M$ directions, respectively (Fig.
2(b)). We indicate the first direct excitonic peak in Fig. 4(b) as D1 at 0.79
eV which originates from the transitions of the parallel bands between
$\Gamma$-$M$ directions with a binding energy of 0.98 eV. After the D1 exciton
in the absorption spectrum, there is a rather flat absorption region with
several exctionic peaks originates from the same paralel bands between
$\Gamma$-$M$ directions until D2 exciton at 1.46 eV. D2 peak has the largest
oscillator strength in the low energy region which coincides with the near
infrared region whose binding energy is 0.77 eV. Our finite-q BSE simulations
showed that there are two indirect bound exciton, ID1 and ID2, at 0.63 eV and
0.67 with a binding energy of 0.56 and 0.52 eV, respectively. Upon comparison
with the solar flux of terrestrial spectra, it is expected that O-hC monolayer
has strong absorption in the deep and near infrared but indirect emission in
the deep infrared regime.

####  III.2.3 O-Ti monolayer

O-Ti monolayer is the only model which has a direct band gap (1.92 eV at the
GW level) among the four Ti2CO2 Mxene monolayer models studied in this
manuscript. The absorption spectrum shows one large peak, D1, which originates
from direct transitions around $M$ point in the BZ at 1.20 eV and has a
binding energy of 0.72 eV. Comparing with the solar flux of terrestrial
spectra reveals that O-Ti monolayer has strong absorption and emission in the
near infrared region. Being a direct band gap semiconductor with an optical
gap in the infrared region signifies the potential usage of O-Ti monolayer in
the infrared laser applications.

####  III.2.4 O-hTiC monolayer

O-hTiC monolayer is the other indirect Ti2CO2 Mxene monolayer model with a
band gap of 1.74 eV at the GW level where the CBM and VBM are at $M$ and
$\Gamma$ points in the BZ, respectively. The absorption spectrum has one
prominent excitonic peak, D1, at 2.02 eV with a binding energy of 0.73 eV. Our
finite-q BSE simulations revealed the existence of an indirect exciton, ID1,
at 1.15 eV, which has a binding energy of 0.59 eV. Comparison with the solar
flux of terrestrial spectra reveals that O-hTiC monolayer has a strong
absorption in the visible region (orange-yellow) and indirect emission in the
infrared region.

##  IV Conclusions

In this manuscript, we present a state-of-the-art first principles analysis,
based on DFT, GW and BSE formalisms, on the electronic and optical properties
of four O-terminated Ti2CO2 MXene monolayers. We show that the electronic band
gap of the monolayer models increase enormously upon inclusion of the GW
correction compared to the LDA values. Using the GW-corrected QP energies and
DFT wave functions, we then solve the BSE in order to investigate the direct
and indirect excitons in these monolayers. We show that the absorption spectra
of the monolayer models drastically redshifted upon inclusion of the electron-
hole interaction due to the large binding energy of the excitons. We also
observe that the binding energy of the indirect excitons are in general lower
than the direct ones, however, they are still the lowest lying excitons in the
absorption spectra of the indirect band gap monolayer models. We find that,
despite some of them being strong absorbers in the visible region, they all
likely are infrared emitters which opens the possibility of their usage in
infrared laser and medical applications. Our findings in this manuscript
emphasize the possible usage of surface termination to tune the optical and
electronic properties of O-terminated monolayer models as well as the
importance of inclusion of the many-body effects for the accurate prediction
of the electronic and optical properties of 2D MXenes in general.

###### Acknowledgements.

This material is based upon work supported by the Air Force Office of
Scientific Research under award number FA9550-19-1-7048. Computing resources
used in this work were provided by the National Center for High Performance
Computing of Turkey (UHeM) under grant number 1007502020. The numerical
calculations reported in this paper were partially performed at TUBITAK
ULAKBIM, High Performance and Grid Computing Center (TRUBA resources). CY
acknowledges the computational support provided by the Computational Shared
Facility at The University of Manchester. FP acknowledges the funding received
from the European Union project MaX Materials design at the eXascale
H2020-INFRAEDI-2018-2020/H2020-INFRAEDI-2018-1, Grant agreement n. 824143.

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