# Lusztig varieties and Macdonald polynomials

Arun Ram email<EMAIL_ADDRESS>

###### Abstract

This paper uses Lusztig varieties to give central elements of the Iwahori-
Hecke algebra corresponding to unipotent conjugacy classes in the finite
Chevalley group $GL_{n}(\mathbb{F}_{q})$. We explain how these central
elements are related to Macdonald polynomials and how this provides a
framework for generalizing integral form and modified Macdonald polynomials to
Lie types other than $GL_{n}$. The key steps are to recognize (a) that
counting points in Lusztig varieties is equivalent to computing traces on the
Hecke algebras, (b) that traces on the Hecke algebra determine elements of the
center of the Hecke algebra, (c) that the Geck-Rouquier basis elements of the
center of the Hecke algebra produce an ‘expansion matrix’, (d) that the
parabolic subalgebras of the Hecke algebra produce a ‘contraction matrix’ and
(e) that the combination ‘expansion-contraction’ is the plethystic
transformation that relates integral form Macdonald polynomials and modified
Macdonald polynomials.

Dedicated to Peter Littelmann

Key words— Macdonald polynomials, Hecke algebras, Lusztig varieties 111AMS
Subject Classifications: Primary 05E05; Secondary 20G99.

##  1 Introduction

###  1.1 Motivation for this paper

Since their introduction by Garsia and Haiman [GH93], the modified Macdonald
polynomials have blossomed into a huge, exciting subfield of algebraic
combinatorics with an extensive literature. A wonderful survey of the history
of the discovery of these amazing objects is in [GR05].

One of the challenges of Macdonald polynomial theory for this author has been
that he did not know (and did not have any good sense for) how to define
modified Macdonald polynomials outside of type A. This paper provides an
answer.

In fact there are three things that need to be generalized to Lie types other
than type $GL_{n}$,

  1. (a)

the integral form Macdonald polynomials $J_{\mu}(x;q,t)$,

  2. (b)

the modified Macdonald polynomials $\tilde{H}_{\mu}[X;q,t]$, and

  3. (c)

the plethystic transformation that relates the integral form and the modified
form.

The route to generalization comes from three connections:

  1. (a)

the monomial expansion of $J_{\mu}(x;q,t)$ counts points of affine Lusztig
varieties,

  2. (b)

the monomial expansion of $\tilde{H}_{\mu}[X;q,t]$ counts points of parabolic
affine Springer fibers, and

  3. (c)

the Lusztig varieties and parabolic Springer fibers are related by expansion-
contractoin.

_Contraction_ is the information of which permutations are in a parabolic
(Young) subgroup – how to ‘contract’ the symmetric group to a Young subgroup.
_Expansion_ is the information of which permutations are in which conjugacy
classes – how a conjugacy ‘expands’ as a sum of permutations; except that we
need this expansion process on the level of the Iwahori-Hecke algebra. The
plethystic transformation is the process of expansion followed by contraction.

###  1.2 Plan of the paper

Section 2 reviews the Iwahori-Hecke algebra and its action on the flag
representation $\mathbf{1}_{B}^{G}$. We explain how counting the points of
Lusztig varieties $Y^{-1}_{BwB}(g)$ and parabolic Springer fibers
$Y^{-1}_{P_{\pi}}(g)$ is captured by traces and central elements in the Hecke
algebra (for an expanded point of view on this counting see [Lu21] and
references there). Using these tools we quickly derive a formula relating the
number of points of the parabolic Springer fiber to the number of points of a
union of Lusztig varieties. This is the geometric source of the expansion-
contraction that plays the essential role in the proof of Theorem 4.1 in
Section 4.

The Hall-Littlewood polynomials are a special case of Macdonald polynomials
(obtained by specializing the variable $q$ to $0$). In Section 3 we review the
connection between integral form Hall-Littlewood polynomials and the number of
points of nilpotent Lusztig varieties which was established in [HR99, Theorem
4.11]. Then we review the connection between modified Hall-Littlewood
polynomials and the number of points in parabolic Springer fibers.

In Section 4 we prove that the integral form Macdonald polynomials and the
modified Macdonald polynomials are related in the same way that the
cardinalities of the Lusztig varieties and the parabolic Springer fibers are
related. This relationship can be elegantly expressed as a product of two
matrices, the expansion matrix and the contraction matrix. The consequence is
that the plethystic transformation which converts integral form Macdonald
polynomials to modified Macdonald polynomials is captured by the expansion-
contraction product.

In section 5 we review the definitions of affine Lusztig varieties and
parabolic affine Springer fibers and the connection between modified Macdonald
polynomials and counting points in nilpotent parabolic affine Springer fibers
given by Mellit [Me17, §5]. This modified Macdonald to parabolic Springer
relationship provides a connection between the monomial expansion of integral
form Macdonald polynomials and counting points of affine Lusztig varieties.

###  1.3 The path to generalization to Lie types other than $GL_{n}$

The expansion-contraction arises very naturally in the setting of the Hecke
algebra $H$ where

  1. (a)

Expansion describes the Geck-Rouquier basis (of the center of $H$) written in
terms of the standard basis of $H$,

  2. (b)

Contraction describes the projection onto a parabolic subalgebra of the Hecke
algebra.

The argument given in this paper (Corollary 2.3 and Proposition 2.4) relating
the cardinalities of Lusztig varieties and parabolic Springer fibers by
expansion-contraction works exactly the same for general Lie types.

The relationship between symmetric functions and the center of $H$ is given by
Wan and Wang in [WW12, Theorem 4.6]. It is an isomorphism that takes

  1. (a)

the monomial symmetric function $m_{\nu}(x)$ to the Geck-Rouquier basis
element $\kappa_{\nu}$, and

  2. (b)

the big Schur function $S_{\lambda}(x;t)$ to the minimal central idempotent
$z^{H}_{\lambda}$.

The integral form Macdonald polynomial $J_{\mu}(x;q,t)$ maps to the element of
the Hecke algebra

$A_{\mu}=\sum_{\nu}\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))\kappa_{\nu},\qquad\hbox{where}\qquad\hbox{$Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})$
is a unipotent affine Lusztig variety.}$

Then expansion-contraction transforms $A_{\mu}$ to a “modified Macdonald
polynomial”, as an element of the Hecke algebra, in the form

$M_{\mu}=\sum_{\pi}\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))1_{P_{\pi}},\qquad\hbox{where}\qquad\hbox{$Y^{-1}_{I_{\pi}}(u_{\mu})$
is a parabolic affine Springer fiber,}$

and $1_{P_{\pi}}$ is the idempotent in the parabolic Hecke algebra
corresponding to the trivial representation (see the proof of Proposition
2.3). The point is that integral form Macdonald polynomials and modified
Macdonald polynomials sit naturally as _elements of the Hecke algebra_ and
capture the cardinalities and relationship between Lusztig varieties and
parabolic Springer fibers.

The only part of this picture that does not extend to Lie types other than
type $GL_{n}$ is the connection between symmetric functions and elements of
the center of the Hecke algebra. Provided one is willing to consider integral
form Macdonald polynomials and modified Macdonald polynomials as elements
$A_{\mu}$ and $M_{\mu}$ in the Hecke algebra then these objects generalize to
all Lie types and serve as useful generating functions for enumerating the
points of affine Lusztig varieties and parabolic affine Springer fibers.

###  1.4 Acknowledgments

I would like to thank Persi Diaconis for drawing me into the work which
resulted in the paper [DRS22]. It was this work that led me to revisit [HR99]
at an opportune moment. I am very grateful to Xuhua He for educating me about
Lusztig varieties. His paper [He23] and discussions with him during his recent
visit to University of Melbourne were very helpful for me.

Finally, it is a pleasure to dedicate this paper to Peter Littelmann who has
been a friend and inspiration for a long time.

##  2 Hecke algebras and $\mathbf{1}_{B}^{G}$

###  2.1 The $(G,H)$-bimodule $\mathbf{1}_{B}^{G}$

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements. Let
$G=GL_{n}(\mathbb{F}_{q})$, let $B$ be the subgroup of upper triangular
matrices and let $W$ be the subgroup of permutation matrices (the symmetric
group $S_{n}$). The Bruhat decomposition is the double coset decomposition

$G=\bigsqcup_{w\in W}BwB.$

(More generally, one can let $G$ be a finite group with a BN-pair and let $W$
be the Weyl group, see [Bou, Ch. IV §2].)

Let $\mathbb{C}G=\hbox{$\mathbb{C}$-span}\\{g\in G\\}$ be the group algebra of
$G$. The Hecke algebra $H$ is the subalgebra of $\mathbb{C}G$ given by

$H=\hbox{$\mathbb{C}$-span}\\{T_{w}\ |\ w\in W\\},\qquad\hbox{where}\qquad
T_{w}=\frac{1}{|B|}\sum_{x\in BwB}x.$

(The Hecke algebra $H$ has unit $T_{1}$; it is a nonunital subalgebra of
$\mathbb{C}G$.) Let $w\in W$ and let $\ell(w)$ denote the length of $w$ and
let $s_{i}$ be a simple reflection in $W$. Then (see [CR81, (67.2)] and/or
[Bou, Ch. IV §2, Ex. 24])

$\frac{|B|}{|BwB|}=q^{-\ell(w)}\qquad\hbox{and}\qquad
T_{s_{i}}T_{w}=\begin{cases}(q-1)T_{w}+qT_{s_{i}w},&\hbox{if
$\ell(s_{i}w)<\ell(w)$,}\\\ T_{s_{i}w},&\hbox{if
$\ell(s_{i}w)>\ell(w)$,}\end{cases}$

and

$T_{w}T_{s_{i}}=\begin{cases}(q-1)T_{w}+qT_{ws_{i}},&\hbox{if
$\ell(ws_{i})<\ell(w)$,}\\\ T_{ws_{i}},&\hbox{if
$\ell(ws_{i})>\ell(w)$.}\end{cases}$ (2.1)

For $g\in G$ let

$v_{g}=\frac{1}{|B|}\sum_{x\in gB}x,\qquad\hbox{and
define}\qquad\mathbf{1}_{B}^{G}=\hbox{$\mathbb{C}$-span}\\{v_{g}\ |\ g\in
G\\}.$ (2.2)

Then $\mathrm{dim}(\mathbf{1}_{B}^{G})=|G/B|$.

$\mathbb{C}G$ acts by left multiplication on $\mathbf{1}_{B}^{G}$ and $H$ acts
by right multiplication on $\mathbf{1}_{B}^{G}$.

As a $G$-module $\mathbf{1}_{B}^{G}$ is isomorphic to the trivial
representation of $B$ induced to $G$, and $H$ is the centralizer algebra,

$\mathbf{1}_{B}^{G}\cong\mathrm{Ind}_{B}^{G}(triv)\qquad\hbox{and}\qquad
H\cong\mathrm{End}_{G}(\mathbf{1}_{B}^{G}).$

Let $\hat{H}$ be an index set for the irreducible $H$-modules. By the
centralizer theorem (see [HR04, Theorem 5.4]), as $(G,H)$-bimodules,

$\mathbf{1}_{B}^{G}\cong\bigoplus_{\lambda\in\hat{H}}G^{\lambda}\otimes
H^{\lambda},$ (2.3)

where the sum is over an index set for irreducible representations of $H$,
$G^{\lambda}$ is an irreducible $G$-module indexed by $\lambda$ and
$H^{\lambda}$ is the irreducible $H$-module indexed by $\lambda$. The
$G^{\lambda}$ are the _unipotent representations_ of $G$.

The irreducible characters of $H$ are the functions

$\chi^{\lambda}_{H}\colon H\to\mathbb{C}\qquad\hbox{given
by}\qquad\chi^{\lambda}_{H}(T_{w})=\mathrm{tr}(T_{w},H^{\lambda}).$

The irreducible unipotent characters of $G$ are the functions

$\chi^{\lambda}_{G}\colon\mathbb{C}G\to\mathbb{C}\qquad\hbox{given
by}\qquad\chi^{\lambda}_{G}(g)=\mathrm{tr}(g,G^{\lambda}).$

The decomposition in (2.3) gives that if $g\in G$ and $w\in W$ then

$\mathrm{tr}(gT_{w},\mathbf{1}_{B}^{G})=\sum_{\lambda\in\hat{H}}\chi^{\lambda}_{G}(g)\chi^{\lambda}_{H}(T_{w}).$
(2.4)

Define an inner product $\langle,\rangle_{H}\colon H\otimes H\to\mathbb{C}$ by

$\langle
h_{1},h_{2}\rangle_{H}=\mathrm{tr}(h_{1}h_{2},\mathbf{1}_{B}^{G}),\qquad\hbox{for
$h_{1},h_{2}\in H$.}$

The basis

$\Big{\\{}q^{-\ell(w)}T_{w^{-1}}\ |\ w\in W\Big{\\}}\qquad\hbox{is the dual
basis to}\quad\\{T_{w}\ |\ w\in W\\},$

with respect to $\langle,\rangle_{H}$ (see [CR81, (11.30)(iii)]).

###  2.2 Bases of the center of $H$

Since $\mathbb{C}G$ is a semisimple algebra, then $\mathbf{1}_{B}^{G}$ is a
semisimple $G$-module and $H=\mathrm{End}_{G}(\mathbf{1}_{B}^{G})$ is a
semisimple algebra. The center of $H$ is

$Z(H)=\\{z\in H\ |\ \hbox{if $h\in H$ then $zh=hz$}\\}.$

As in (2.3), let $\hat{H}$ be an index set for the irreducible $H$-modules.
The _minimal idempotent_ basis of $Z(H)$ (see [CR81, (68.29)] or [HLR, (1.6)])
is

$\\{z_{\lambda}^{H}\ |\ \lambda\in\hat{H}\\},\qquad\hbox{where}\quad
z_{\lambda}^{H}=\frac{\chi^{\lambda}_{G}(1)}{|G/B|}\sum_{w\in
W}\chi^{\lambda}_{H}(T_{w})q^{-\ell(w)}T_{w^{-1}}.$ (2.5)

Let $\mathcal{W}$ be an index set for the conjugacy classes of $W$. The
_conjugacy class basis_ , or _Geck-Rouquier basis_ , of $Z(H)$ (see [GP00,
Cor. 8.2.4]) is

$\\{\kappa_{\nu}\ |\ \nu\in\mathcal{W}\\}\quad\hbox{given
by}\qquad\kappa_{\nu}=\sum_{w\in
W}\kappa_{\nu,w}q^{-\ell(w)}T_{w}\qquad\hbox{satisfying}\qquad\kappa_{\nu,\gamma_{\mu}}=\delta_{\nu\mu},$
(2.6)

whenever $\gamma_{\mu}$ is a minimal length of the conjugacy class
$\mathcal{W}_{\mu}$. The condition $\kappa_{\nu,\gamma_{\mu}}=\delta_{\nu\mu}$
determines $\kappa_{\nu,w}$ for $w$ that are minimal length in their conjugacy
class in $W$ and the remaining $\kappa_{\nu,w}$ are forced by the condition
that $\kappa\in Z(H)$. The transition matrix between the minimal central
idempotents and the conjugacy class basis is the character table of the Hecke
algebra,

$z_{\lambda}^{H}=\sum_{\mu\in\mathcal{W}}\chi^{\lambda}_{H}(T_{\gamma_{\nu}})\kappa_{\nu},\qquad\hbox{where}\qquad\begin{array}[]{l}\hbox{$\gamma_{\mu}$
is a minimal length element}\\\ \hbox{ in the conjugacy class
$\mathcal{W}_{\mu}$.}\end{array}$

###  2.3 Conjugacy classes, Schubert cells and Lusztig varieties

Let $\mathcal{C}$ be an index set for the conjugacy classes of $G$.

$G=\bigsqcup_{\mu\in\mathcal{C}}\mathcal{C}_{\mu}\qquad\hbox{and}\qquad
G=\bigsqcup_{w\in W}BwB.$

For $g\in G$ and $w\in W$,

$\hbox{the \emph{Lusztig variety} is}\qquad Y_{BwB}^{-1}(g)=\\{yB\in G/B\ |\
y^{-1}gy\in BwB\\}.$

The following proposition shows that the number of points of $Y_{BwB}^{-1}(g)$
is related to the size of the intersection of the conjugacy class of $g$ with
the Schubert cell $Bw^{-1}B$.

######  Proposition 2.1.

Let $g\in G$ and let $\mathcal{C}_{g}$ be the conjugacy class of $g$.

Let $w\in W$. Then

$\displaystyle\mathrm{Card}(Y^{-1}_{BwB}(g^{-1}))$
$\displaystyle=\mathrm{tr}(gT_{w},\mathbf{1}_{B}^{G})=\frac{|G/B|}{|\mathcal{C}_{g}|}\mathrm{Card}(\mathcal{C}_{g}\cap
Bw^{-1}B)=\sum_{\lambda\in\hat{H}}\chi^{\lambda}_{G}(g)\chi^{\lambda}_{H}(T_{w})$

Using notations as in (2.5) and (2.6), the element of $Z(H)$ given by

$A_{g}=\sum_{w\in
W}\mathrm{tr}(gT_{w},\mathbf{1}_{B}^{G})q^{-\ell(w)}T_{w^{-1}}=\sum_{\nu\in\mathcal{W}}\mathrm{tr}(gT_{\gamma_{\nu}},\mathbf{1}_{B}^{G})\kappa_{\nu}=|G/B|\sum_{\lambda\in\hat{H}}\frac{\chi^{\lambda}_{G}(g)}{\chi^{\lambda}_{G}(1)}z^{H}_{\lambda}$

acts on $\mathbf{1}_{B}^{G}$ the same way as the element of $Z(\mathbb{C}G)$
given by $\frac{|G/B|}{|\mathcal{C}_{g}|}C_{g}$, where

$C_{g}=\sum_{x\in\mathcal{C}_{g}}x.$

###### Proof.

By definition $T_{1}=\frac{1}{|B|}\sum_{x\in B}x$. If $b\in B$ then
$bT_{1}=T_{1}b=T_{1}$ and $T_{1}^{2}=T_{1}$. If $g\in BwB$ and $g=b_{1}wb_{2}$
with $b_{1},b_{2}\in B$ then

$\displaystyle T_{1}gT_{1}$
$\displaystyle=T_{1}b_{1}wb_{2}T_{1}=T_{1}wT_{1}=\frac{1}{|B|^{2}}\sum_{b_{1},b_{2}\in
B}b_{1}wb_{2}$ $\displaystyle=\frac{1}{|B|^{2}}\frac{|B|^{2}}{|BwB|}\sum_{x\in
BwB}x=\frac{|B|}{|BwB|}\frac{1}{|B|}\sum_{x\in
BwB}x=\frac{|B|}{|BwB|}T_{w}=q^{-\ell(w)}T_{w}.$ (2.7)

(a) Let $z_{\lambda}^{G}$ be the minimal central idempotent in
$Z(\mathbb{C}G)$ which acts on $G^{\lambda}$ by the identity. Then
$\frac{|G/B|}{|\mathcal{C}_{g}|}C_{g}$ acts on $\mathbf{1}_{B}^{G}$ the same
way as

$\frac{|G/B|}{|\mathcal{C}_{g}|}\sum_{\lambda\in\hat{H}}\frac{\chi^{\lambda}_{G}(C_{g})}{\chi_{G}^{\lambda}(1)}z^{G}_{\lambda}=\frac{|G/B|}{|\mathcal{C}_{g}|}\sum_{\lambda\in\hat{H}}|\mathcal{C}_{g}|\frac{\chi^{\lambda}_{G}(g)}{\chi_{G}^{\lambda}(1)}z^{G}_{\lambda}$

and the same way as

$\displaystyle A_{g}$
$\displaystyle=\frac{|G/B|}{|\mathcal{C}_{g}|}\sum_{\lambda\in\hat{H}}\frac{|\mathcal{C}_{g}|}{\chi^{\lambda}_{G}(1)}\chi^{\lambda}_{G}(g)z^{H}_{\lambda}$
$\displaystyle=|G/B|\sum_{\lambda\in\hat{H}}\frac{\chi^{\lambda}_{G}(g)}{\chi^{\lambda}_{G}(1)}\frac{1}{|G/B|}\sum_{w\in
W}\chi^{\lambda}_{G}(1)\chi^{\lambda}_{H}(T_{w})q^{-\ell(w)}T_{w^{-1}}$
$\displaystyle=\sum_{w\in
W}\Big{(}\sum_{\lambda\in\hat{H}}\chi^{\lambda}_{G}(g)\chi^{\lambda}_{H}(T_{w})\Big{)}q^{-\ell(w)}T_{w^{-1}}$
$\displaystyle=\sum_{w\in
W}\mathrm{tr}(gT_{w},\mathbf{1}_{B}^{G})q^{-\ell(w)}T_{w^{-1}},$ (2.8)

where the second equality uses (2.5) and the last equality follows from (2.4).

Using that $C_{g}$ is central in $\mathbb{C}G$ gives
$C_{g}T_{1}=C_{g}T_{1}^{2}=T_{1}C_{g}T_{1}$ so that

$\displaystyle A_{g}=\frac{|G/B|}{|\mathcal{C}_{g}|}C_{g}T_{1}$
$\displaystyle=\frac{|G/B|}{|\mathcal{C}_{g}|}\sum_{x\in\mathcal{C}_{g}}T_{1}xT_{1}=\sum_{w\in
W}\frac{|G/B|}{|\mathcal{C}_{g}|}\mathrm{Card}(\mathcal{C}_{g}\cap
Bw^{-1}B)q^{-\ell(w)}T_{w^{-1}},$ (2.9)

where the last equality follows from (2.7). Comparing coefficients of $T_{w}$
in (2.8) and (2.9) gives

$\frac{|G/B|}{|\mathcal{C}_{g}|}\mathrm{Card}(\mathcal{C}_{g}\cap
Bw^{-1}B)=\mathrm{tr}(gT_{w},\mathbf{1}_{B}^{G})=\sum_{\lambda\in\hat{H}}\chi^{\lambda}_{G}(g)\chi^{\lambda}_{H}(T_{w})\,.$

For $h\in\mathbf{1}_{B}^{G}$, let $h|_{v_{g}}$ denote the coefficient of
$v_{g}$ when $h$ is expanded in the basis $\\{v_{g}\ |\ gB\in G/B\\}$ given in
(2.2). Then

$\displaystyle\mathrm{tr}($ $\displaystyle
gT_{w},\mathbf{1}_{B}^{G})=\sum_{yB\in
G/B}gv_{y}T_{w}\big{|}_{v_{y}}=\sum_{yB\in
G/B}v_{y}T_{w}\big{|}_{v_{g^{-1}y}}=\sum_{yB\in G/B}\sum_{zB\in G/B\atop zB\in
yBwB}v_{z}\big{|}_{v_{g^{-1}y}}$ $\displaystyle=\\#\\{yB\in G/B\ |\
g^{-1}yB\in yBwB\\}=\\#\\{yB\in G/B\ |\ y^{-1}g^{-1}y\in
BwB\\}=\\#Y^{-1}_{BwB}(g^{-1}).$

∎

A _trace on $H$_ is a linear function $\chi\colon H\to\mathbb{C}$ such that

if $h_{1},h_{2}\in H$ then $\chi(h_{1}h_{2})=\chi(h_{2}h_{1})$.

Using (2.1), if $\chi\colon H\to\mathbb{C}$ is a trace then

$\chi(T_{s_{i}w})=\chi(T_{s_{i}}T_{w})=\chi(T_{w}T_{s_{i}})=\chi(T_{ws_{i}}),\qquad\hbox{if
$\ell(s_{i}ws_{i})=\ell(w)$ and $\ell(s_{i}w)=\ell(w)+1$. and}$
$\chi(T_{s_{i}ws_{i}})=\chi(T_{s_{i}}T_{w}T_{s_{i}})=\chi(T^{2}_{s_{i}}T_{w})=q\chi(T_{w})+(q-1)\chi(T_{s_{i}w}),\qquad\hbox{if
$\ell(s_{i}ws_{i})=\ell(w)+2$.}$

Since
$\mathrm{Card}(Y^{-1}_{BwB}(g))=\mathrm{tr}(g^{-1}T_{w},\mathbf{1}_{B}^{G})$
is a trace (as a function on $H$) then these trace relations imply the
following Corollary. Corollary 2.2 is also a consequence of the slightly more
refined statement about the structure of Lusztig varieties stated in [He23,
§4.4] (with a reference to [DL76, proof of Theorem 1.6] for the proof).

######  Corollary 2.2.

Let $g\in G$. Let $w\in W$ and $i\in\\{1,\ldots,n-1\\}$ such that
$\ell(s_{i}w)=\ell(w)+1$. Then

$\displaystyle\mathrm{Card}(Y^{-1}_{Bs_{i}wB}(g))$
$\displaystyle=\mathrm{Card}(Y^{-1}_{Bws_{i}B}(g)),$ if
$\ell(s_{i}w)=\ell(ws_{i})$, and
$\displaystyle\mathrm{Card}(Y^{-1}_{Bs_{i}ws_{i}B}(g))$
$\displaystyle=q\mathrm{Card}(Y^{-1}_{Bs_{i}wB}(g))+(q-1)\mathrm{Card}(Y^{-1}_{BwB}(g)),$
if $\ell(s_{i}ws_{i})=\ell(w)+2$.

###  2.4 Parabolic Springer fibers

Let $\pi=(\pi_{1},\ldots,\pi_{\ell})$ with
$\pi_{1},\ldots,\pi_{\ell}\in\mathbb{Z}_{>0}$ and
$\pi_{1}+\cdots+\pi_{\ell}=n$. Let $P_{\pi}$ be the parabolic subgroup of $G$
consisting of block upper triangular matrices with block sizes
$\pi_{1},\ldots,\pi_{\ell}$. Let $W_{\pi}$ be the subgroup of $W=S_{n}$ given
by

$W_{\pi}=S_{\pi_{1}}\times\cdots\times S_{\pi_{\ell}}\qquad\hbox{so
that}\qquad P_{\pi}=\bigsqcup_{w\in W_{\pi}}BwB.$

Let

$W_{\pi}(q)=\sum_{w\in W_{\pi}}q^{\ell(w)}\qquad\hbox{so
that}\qquad\mathrm{Card}(P_{\pi}/B)=W_{\pi}(q).$

Let $g\in G$. The _$\pi$ -parabolic Springer fiber over $g$_ is

$Y^{-1}_{P_{\pi}}(g)=\\#\\{yP_{\pi}\in G/P_{\pi}\ |\ y^{-1}gy\in P_{\pi}\\}.$
(2.10)

The following Proposition counts the number of points of $Y^{-1}_{P_{\pi}}(g)$
in terms of the sizes of the Lusztig varieties.

######  Proposition 2.3.

$\mathrm{Card}(Y^{-1}_{P_{\pi}}(g))=\frac{1}{W_{\pi}(q)}\sum_{w\in
W_{\pi}}\mathrm{Card}(Y^{-1}_{BwB}(g)).$

###### Proof.

Let

$W_{\pi}(q)=\sum_{w\in W_{\pi}}q^{\ell(w)}\qquad\hbox{and let}\qquad
1_{P_{\pi}}=\frac{1}{W_{\pi}(q)}\sum_{w\in W_{\pi}}T_{w},$

which is an idempotent in the Hecke algebra $H$. As a $G$-module

$\mathbf{1}_{P_{\pi}}^{G}=\mathbf{1}_{B}^{G}\cdot
1_{P_{\pi}}\cong\mathrm{Ind}_{P_{\pi}}^{G}(\mathrm{triv}),$

the trivial representation of $P_{\pi}$ induced to $G$. Then the number of
points of $Y^{-1}_{P_{\pi}}(g)$ is given by a trace:

$\displaystyle\mathrm{Card}(Y^{-1}_{P_{\pi}}(g))$
$\displaystyle=\\#\\{yP_{\pi}\ |\
gyP_{\pi}=yP_{\pi}\\}=\mathrm{tr}(g,\mathbf{1}_{P_{\pi}}^{G})=\mathrm{tr}(g,\mathbf{1}_{B}^{G}\cdot
1_{P_{\pi}})$ $\displaystyle=\frac{1}{W_{\pi}(q)}\sum_{w\in
W_{\pi}}\mathrm{tr}(gT_{w},\mathbf{1}_{B}^{G})=\frac{1}{W_{\pi}(q)}\sum_{w\in
W_{\pi}}\mathrm{Card}(Y^{-1}_{BwB}(g)).$

∎

##  3 Counting in $GL_{n}(\mathbb{F}_{q})$

###  3.1 Macdonald polynomials

Fix $n\in\mathbb{Z}_{>0}$. A _partition of $n$_ is a sequence
$\lambda=(\lambda_{1},\ldots,\lambda_{\ell})$ of positive integers with
$\lambda_{1}\geq\ldots\geq\lambda_{\ell}>0$ and
$\lambda_{1}+\cdots+\lambda_{n}$. For
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$, define

$\ell(\lambda)=\ell\qquad\qquad\hbox{and}\qquad
n(\lambda)=\sum_{i=1}^{\ell}(i-1)\lambda_{i}.$

Let

  1. $J_{\mu}(x;q,t)$ be the integral form Macdonald polynomials [Mac, Ch. VI (8.3)],

  2. $S_{\lambda}(x;t)$ the Big Schur functions [Mac, Ch. III (4.5)] and

  3. $m_{\nu}$ the monomial symmetric functions.

Define $K_{\lambda\nu}(q,t)$, $a_{\mu\nu}(q,t)$ and $L_{\nu\lambda}(t)$ by

$J_{\mu}(x;q,t)=\sum_{\lambda}K_{\lambda\mu}(q,t)S_{\lambda}(x;t),\qquad\hbox{and}\qquad
J_{\mu}(x;q,t)=\sum_{\nu}a_{\mu\nu}(q,t)(1-t)^{\ell(\nu)}m_{\nu},$

and

$S_{\lambda}(x;t)=\sum_{\nu}L_{\lambda\nu}(t)(1-t)^{\ell(\nu)}m_{\nu},\qquad\hbox{so
that}\qquad
a_{\mu\nu}(q,t)=\sum_{\lambda}K_{\lambda\mu}(q,t)L_{\lambda\nu}(t).$ (3.1)

The Schur functions $s_{\lambda}$ and the modified Macdonald polynomials
$\widetilde{H}_{\mu}[X;q,t]$ are given by

$s_{\lambda}=\sum_{\pi}K_{\lambda\pi}(0,1)m_{\pi}.$ (3.2)
$\begin{array}[]{l}\displaystyle{\widetilde{H}_{\mu}[X;q,t]=\sum_{\lambda}t^{n(\lambda)}K_{\lambda\mu}(q,t^{-1})s_{\lambda},}\\\
\displaystyle{\widetilde{H}_{\mu}[X;q,t]=\sum_{\pi}b_{\mu\pi}(q,t)m_{\pi},}\end{array}\qquad\hbox{so
that}\qquad
b_{\mu\nu}(q,t)=\sum_{\lambda}t^{n(\lambda)}K_{\lambda\mu}(q,t^{-1})K_{\lambda\pi}(0,1).$

We follow [Mac, Ch. VI] in using nonplethystic notation for the variables in
$J_{\mu}(x;q,t)$ and [GH96] and [Me17] in using plethystic notation for the
variables in $\widetilde{H}_{\mu}[X;q,t]$.

Another way to state the relation between $\widetilde{H}_{\mu}$ and $J_{\mu}$
is via the plethystic transformation which has the effect of changing
$S_{\lambda}(x;t)$ to $s_{\lambda}(x)$. As in [GH96, (11)],

$\widetilde{H}_{\lambda}[X;q,t]=t^{n(\lambda)}J_{\lambda}\Big{[}\frac{X}{1-t^{-1}};q,t^{-1}\Big{]}.$
(3.3)

From [GH96, Cor. 3.2] or [Mac, (8.14) and (8.15)],

$\widetilde{H}_{\lambda}[X;q,t]=\widetilde{H}_{\lambda^{\prime}}[X;t,q],$
(3.4)

where $\lambda^{\prime}$ is the conjugate partition to $\lambda$.

###  3.2 Counting points of Lusztig varieties in $GL_{n}(\mathbb{F}_{q})$

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements. Let $B$ be the
subgroup of upper triangular matrices in $G=GL_{n}(\mathbb{F}_{q})$ and let
$H$ be the Hecke algebra for $B\subseteq G$.

Let $\lambda,\mu,\nu$ be partitions of $n$. Let $\chi^{\lambda}_{G}$ be the
character of the irreducible unipotent $GL_{n}(\mathbb{F}_{q})$-representation
$G^{\lambda}$ and let $u_{\mu}$ be a unipotent element in
$GL_{n}(\mathbb{F}_{q})$ with Jordan form corresponding to $\mu$. From [Lu81,
(2.2)] or [HR99, Theorem 4.9(c)],

$\chi^{\lambda}_{G}(u_{\mu})=q^{n(\mu)}K_{\lambda\mu}(0,q^{-1}).$ (3.5)

This identity provides a representation theoretic viewpoint on (a
specialization) of the coefficients $K_{\lambda\mu}(q,t)$ which appear in
(3.1).

With respect to the inner product $\langle,\rangle_{0,t}$ of [Mac, Ch. III],
the Schur functions $s_{\lambda}$ and the Big Schur functions
$S_{\lambda}=S_{\lambda}(x;t)$ are dual bases. The dual basis to the monomial
symmetric functions $m_{\mu}$ is denoted $q_{\mu}=q_{\mu}(x;t)$ in [Mac, (4.8)
and (4.10)]. In formulas, $\langle
s_{\lambda},S_{\mu}\rangle_{0,t}=\delta_{\lambda\mu}$ and $\langle
q_{\nu},m_{\mu}\rangle_{0,t}=\delta_{\nu\mu}.$ Thus

$L_{\lambda\nu}(t)(1-t)^{\ell(\nu)}=\langle
q_{\nu},S_{\lambda}\rangle_{0,t}\qquad\hbox{which
gives}\qquad\frac{1}{(1-t)^{\ell(\nu)}}q_{\nu}(x;t)=\sum_{\lambda}L_{\lambda\nu}(t)s_{\lambda}(x).$

Let $\lambda,\nu$ be partitions of $n$. Let $\chi^{\lambda}_{H}$ be the
character of the irreducible $H$-representation $H^{\lambda}$. Let
$\gamma_{\nu}$ be a minimal length element of the conjugacy class in $W$ of
permutations of cycle type $\nu$. By [Ra91, Th. 4.14],

$\frac{q^{n-\ell(\nu)}}{(q-1)^{\ell(\nu)}}q_{\nu}(x;q^{-1})=\sum_{\lambda}\chi^{\lambda}_{H}(T_{\gamma_{\nu}^{-1}})s_{\lambda}(x)$

so that

$\chi^{\lambda}_{H}(T_{\gamma_{\nu}^{-1}})=q^{n-\ell(\nu)}L_{\lambda\nu}(q^{-1}).$
(3.6)

This identity provides a representation theoretic viewpoint on the
coefficients $L_{\lambda\nu}(t)$ which appear in (3.1).

The following Theorem is a reformulation of the first displayed equation in
the proof of [HR99, Theorem 4.11]. It provides a geometric viewpoint for a
specialization of the coefficients $a_{\mu\nu}(q,t)$ which appear in (3.1).

######  Theorem 3.1.

Let $\mu,\nu$ be partitions of $n$. Let $u_{\mu}$ be a unipotent element in
$GL_{n}(\mathbb{F}_{q})$ with Jordan form corresponding to $\mu$ and let
$\gamma_{\nu}$ be a minimal length element of the conjugacy class in $W$ of
permutations of cycle type $\nu$. Then

$q^{n(\mu)+n-\ell(\nu)}a_{\mu\nu}(0,q^{-1})=\mathrm{Card}(Y^{-1}_{B\gamma_{\nu}B}(u_{\mu})),$

the number of points of the Lusztig variety $Y^{-1}_{B\gamma_{\nu}B}(u_{\mu})$
over $\mathbb{F}_{q}$.

###### Proof.

By Proposition 2.1, (3.5), (3.6) and the last identity in (3.1),

$\displaystyle\mathrm{Card}(Y^{-1}_{B\gamma_{\nu}B}(u_{\mu}))$
$\displaystyle=\sum_{\lambda}\chi^{\lambda}_{G}(u_{\mu})\chi^{\lambda}_{H}(T_{\gamma_{\nu}^{-1}})=\sum_{\lambda}q^{n(\mu)}K_{\lambda\mu}(0,q^{-1})L_{\lambda\nu}(q^{-1})q^{n-\ell(\nu)}$
$\displaystyle=q^{n(\mu)+n-\ell(\nu)}a_{\mu\nu}(0,q^{-1}),$

∎

###  3.3 Modified Hall-Littlewoods and parabolic Springer fibers

Recall from (3.1) and (3.2) that the $a_{\mu\nu}(q,t)$ give the expansion of
integral form Macdonald polynomials in monomial symmetric functions and the
$b_{\mu\pi}(q,t)$ specify the transition matrix between the modified Macdonald
polynomials and the monomial symmetric functions,

$J_{\mu}(x;q,t)=\sum_{\nu}a_{\mu\nu}(q,t)(1-t)^{\ell(\nu)}m_{\nu}\qquad\hbox{and}\qquad\widetilde{H}_{\mu}[X;q,t]=\sum_{\pi}b_{\mu\pi}(q,t)m_{\pi}.$

The corresponding Hall-Littlewood polynomials are the specializations
$J_{\mu}(x;0,t)$ and $\widetilde{H}_{\mu}[X;0,t]$.

An elegant proof of the following result is found in [Me17, Theorem 2.12 and
Cor. 2.13]. Alternative references are [LLT95, (9)] and [HS79, Cor. 8.7].
Although the result can be deduced fairly easily from identities in [Mac] (see
[LLT95, (9)], [HR99, Theorem 4.9(a)], [Mac, Ch. IV §4] and [Mac, Ch. IV §4 Ex.
1]) getting the necessary variable specializations and conjugate partitions
organized cleanly is painful enough that it is convenient to stick with the
clean, direct, proof found in [Me17, Theorem 2.12 and Cor. 2.13].

######  Theorem 3.2.

Let $\mu$ and $\pi$ be partitions of $n$. Let $u_{\mu}\in
GL_{n}(\mathbb{F}_{q})$ be a unipotent element in Jordan normal form with
Jordan block sizes given by the partition $\mu$ and let $P_{\pi}\subseteq
GL_{n}(\mathbb{F}_{q})$ be the standard parabolic subgroup of block upper
triangular matrices with blocks sizes determined by the partition $\pi$. Let
$Y^{-1}_{P_{\pi}}(u_{\mu})$ be the $\pi$-parabolic Springer fiber over
$u_{\mu}$ defined in (2.10). Then

$b_{\mu\pi}(0,q)=\mathrm{Card}(Y^{-1}_{P_{\pi}}(u_{\mu}^{-1})).$

##  4 Expansion-contraction

Let $n\in\mathbb{Z}_{>0}$. As in (2.6), let $\mathcal{W}$ be an index set for
the conjugacy classes of $W$ (for $W=S_{n}$ the set $\mathcal{W}$ is naturally
identified with the set of partitions of $n$).

The _expansion matrix_ is $\kappa=(\kappa_{\nu,w})$ (rows indexed by
$\mu\in\mathcal{W}$ and columns indexed by $w\in W$) given by

  1. (a)

$\kappa_{\nu,\gamma_{\mu}}=\delta_{\mu\nu}$, if $\gamma_{\mu}$ is minimal
length in the conjugacy class $\mathcal{W}_{\mu}$,

  2. (b)

$\kappa_{\nu,s_{i}w}=\kappa_{\nu,ws_{i}}$ if $\ell(s_{i}w)=\ell(ws_{i})$ and
$\ell(s_{i}w)=\ell(w)+1$,

  3. (c)

$\kappa_{\nu,s_{i}ws_{i}}=q\kappa_{\nu,w}+(q-1)\kappa_{\nu,ws_{i}}$ if
$\ell(s_{i}ws_{i})=\ell(w)+2$.

Algebraically, the matrix $\kappa$ contains the coefficient of the Geck-
Rouquier basis of $Z(H)$ when expanded in terms of the basis
$\\{q^{-\ell(w)}T_{w^{-1}}\ |\ w\in W\\}$ of $H$. Geometrically, the expansion
matrix is related to the trace identities for Lusztig varieties in Corollary
2.2.

With notations as in Section 2.4, the _contraction matrix_ is $C=(C_{w,\pi})$
given by

$C_{w,\pi}=\begin{cases}1,&\hbox{if $w\in W_{\pi}$,}\\\ 0,&\hbox{if $w\not\in
W_{\pi}$,}\end{cases}\qquad\hbox{where}\qquad
W_{\pi}=S_{\pi_{1}}\times\cdots\times S_{\pi_{\ell}}$

is a parabolic subgroup of $W$ (a Young subgroup of $S_{n}$) corresponding to
the partition $\pi=(\pi_{1},\ldots,\pi_{\ell})$. Algebraically, the
contraction matrix captures the idempotent $\mathbf{1}_{\pi}\in H$ that
projects $\mathbf{1}_{B}^{G}\to\mathbf{1}_{P_{\pi}}^{G}$. Geometrically, the
contraction describes the relation between Lusztig varieties and parabolic
Springer fibers that appears in Proposition 2.3.

###  4.1 Expansion-contraction and Macdonald polynomials

Recall from (3.1) and (3.2) that the $a_{\mu\nu}(q,t)$ give the expansion of
integral form Macdonald polynomials in monomial symmetric functions and the
$b_{\mu\pi}(q,t)$ specify the transition matrix between the modified Macdonald
polynomials and the monomial symmetric functions,

$J_{\mu}(x;q,t)=\sum_{\nu}a_{\mu\nu}(q,t)(1-t)^{\ell(\nu)}m_{\nu}\qquad\hbox{and}\qquad\widetilde{H}_{\mu}[X;q,t]=\sum_{\pi}b_{\mu\pi}(q,t)m_{\pi},$

For $k\in\mathbb{Z}_{>0}$ let

$[k]=1+q+\cdots+q^{k-1},\quad[k]!=[k][k-1]\cdots[2][1],\qquad\hbox{and
let}\qquad[\pi]!=[\pi_{1}]!\cdots[\pi_{\ell}]!.$

The following Theorem tells us that, up to normalizations, the
$b_{\mu\pi}(t,q)$ are obtained from the $a_{\mu\nu}(t,q^{-1})$ by multiplying
the the expansion matrix and the contraction matrix.

######  Theorem 4.1.

Let $\mu,\pi$ be partitions of $n$. Then

$\sum_{\nu\vdash n}\sum_{w\in
S_{n}}q^{n(\mu)}a_{\mu\nu}(t,q^{-1})q^{n-\ell(\nu)}\kappa_{\nu,w}C_{w,\pi}\frac{1}{[\pi]!}=b_{\mu\pi}(t,q).$

###### Proof.

Let $A(t,q^{-1})$, $b(t,q)$, $D$ and $M$ be the matrices given by

$\displaystyle A(t,q^{-1})=(q^{n(\mu)}a_{\mu\nu}(t,q^{-1})),\qquad\qquad
b(t,q)=(b_{\mu\pi}(t,q)),$ $D=\mathrm{diag}(d_{\pi})$ is the diagonal matrix
with diagonal entries $d_{\pi}=\frac{1}{[\pi]!}$, and
$M=\mathrm{diag}(q^{n-\ell(\nu)})$ is the diagonal matrix with diagonal
entries $q^{n-\ell(\nu)}$.

Using the notations in (3.2) and (3.1), define matrices

$K(0,1)=(K_{\lambda\mu}(0,1)),\qquad
F=K(0,1)^{-1}=(F_{\pi\lambda})\qquad\hbox{and}\qquad
L=(L_{\lambda\nu}(q^{-1}))$

so that

$m_{\pi}=\sum_{\lambda}F_{\pi\lambda}s_{\lambda}\qquad\hbox{and}\qquad
s_{\lambda}[X(1-q^{-1})]=S_{\lambda}(q^{-1})=\sum_{\nu}L_{\lambda\nu}(q^{-1})(1-q^{-1})m_{\nu}[X].$

Using the identities in (3.1), (3.2) and (3.3),

$\displaystyle\sum_{\nu}q^{n(\mu)}a_{\mu\nu}(t,q^{-1})(1-q^{-1})^{\ell(\nu)}m_{\nu}[X]$
$\displaystyle=q^{n(\mu)}J_{\mu}[X;t,q^{-1}]=\widetilde{H}_{\mu}[X(1-q^{-1});t,q]$
$\displaystyle=\sum_{\pi}b_{\mu\pi}(t,q)m_{\pi}[X(1-q^{-1})]$
$\displaystyle=\sum_{\pi,\lambda}b_{\mu\pi}(t,q)F_{\pi\lambda}s_{\lambda}[X(1-q^{-1})]$
$\displaystyle=\sum_{\pi,\lambda}b_{\mu\pi}(t,q)F_{\pi\lambda}L_{\lambda\nu}(q^{-1})(1-q^{-1})^{\ell(\nu)}m_{\nu}[X].$

Thus

$A(t,q^{-1})=b(t,q)FL.$ (4.1)

Since $\mathrm{Card}(P_{\pi}/B)=[\pi]!$ then Theorem 3.1, Corollary 2.2,
Proposition 2.3, and Theorem 3.2 give

$\displaystyle\sum_{\nu\vdash n}\sum_{w\in
S_{n}}q^{n(\mu)}a_{\mu\nu}(0,q^{-1})q^{n-\ell(\nu)}\kappa_{\nu,w}C_{w,\pi}\frac{1}{[\pi]!}$
$\displaystyle=\sum_{\nu\vdash n}\sum_{w\in
S_{n}}\mathrm{Card}(Y^{-1}_{B\gamma_{\nu}B}(u_{\mu}))\kappa_{\nu,w}C_{w,\pi}\frac{1}{[\pi]!}$
$\displaystyle=\sum_{w\in
S_{n}}\mathrm{Card}(Y^{-1}_{BwB}(u_{\mu}))C_{w,\pi}\frac{1}{[\pi]!}$
$\displaystyle=\mathrm{Card}(Y^{-1}_{P_{\pi}}(u_{\mu}))\mathrm{Card}(P_{\pi}/B)\frac{1}{[\pi]!}$
$\displaystyle=\mathrm{Card}(Y^{-1}_{P_{\pi}}(u_{\mu}))=b_{\mu\pi}(0,q),$

so that

$A(0,q^{-1})M\kappa CD=b(0,q).$

By (4.1),

$A(0,q^{-1})=b(0,q)FL,\qquad\hbox{which gives}\qquad M\kappa CD=(FL)^{-1}.$
(4.2)

Plugging this back into (4.1) gives $A(q,t)M\kappa CD=b(t,q)$, which is
equivalent to the statement in Theorem 4.1. ∎

###  4.2 Expansion-contraction and the plethystic transformation

Let $R=(R_{\nu\pi})$ be the matrix with rows and columns indexed by partitions
of $n$ given by

$m_{\nu}\left[\frac{X}{1-q^{-1}}\right]=\sum_{\pi}R_{\nu\pi}(q)m_{\pi}[X].$

The matrix $R$ is the change of basis matrix for the plethystic
transformation, with respect to the monomial symmetric functions. The
following Corollary says that the matrix $R$ is, up to normalization of rows
and columns, the expansion-contraction. The matrix $R$ is a square matrix, but
the expansion matrix $\kappa$ and the contraction matrix $C$ are not square.

######  Corollary 4.2.

Let $\nu$ and $\pi$ be partitions of $n$. Then

$R_{\nu\pi}(q)=\sum_{w\in
S_{n}}q^{n-\ell(\nu)}\kappa_{\nu,w}C_{w,\pi}\frac{1}{[\pi]!}.$

###### Proof.

Use the matrix notations from the proof of Theorem 4.1. Since

$\displaystyle m_{\pi}[X(1-q^{-1})]$
$\displaystyle=\sum_{\lambda}F_{\pi\lambda}s_{\lambda}[X(1-q^{-1})]\quad\hbox{and}\quad
s_{\lambda}[X(1-q^{-1})]=\sum_{\nu}L_{\lambda\nu}(1-q^{-1})^{\ell(\nu)}m_{\nu}[X].$

then

$R^{-1}=FL\qquad\hbox{so that}\qquad R=(FL)^{-1}=M\kappa CD,$

by (4.2). ∎

##  5 Counting points in affine Springer fibers and Lusztig varieties

Let $B(\mathbb{F}_{q})$ be the subgroup of $GL_{n}(\mathbb{F}_{q})$ of upper
triangular matrices and for a partition $\pi=(\pi_{1},\ldots,\pi_{\ell})$ of
$n$ let

$P_{\pi}(\mathbb{F}_{q})\subseteq GL_{n}(\mathbb{F}_{q})\qquad\hbox{be the
subgroup of $GL_{n}(\mathbb{F}_{q})$ of}\qquad\begin{array}[]{c}\hbox{block
upper triangular matrices}\\\ \hbox{with block sizes
$\pi_{1},\ldots,\pi_{\ell}$.}\end{array}$

Then $B(\mathbb{F}_{q})=P_{(1^{n})}(\mathbb{F}_{q})$. Define $G$, $K$,
$I_{\pi}$ and $I$ by

$\begin{matrix}\widetilde{G}=GL_{n}(\mathbb{F}_{q}((\epsilon)))\\\ \cup|\\\
K=GL_{n}(\mathbb{F}_{q}[[\epsilon]])&\stackrel{{\scriptstyle\epsilon=0}}{{\longrightarrow}}&GL_{n}(\mathbb{F}_{q})\\\
\cup|&&\cup|\\\ I_{\pi}=\\{g\in K\ |\ g(0)\in
P_{\pi}(\mathbb{F}_{q})\\}&\stackrel{{\scriptstyle\epsilon=0}}{{\longrightarrow}}&P_{\pi}(\mathbb{F}_{q})\\\
\cup|&&\cup|\\\ I=\\{g\in K\ |\ g(0)\in
B(\mathbb{F}_{q})\\}&\stackrel{{\scriptstyle\epsilon=0}}{{\longrightarrow}}&B(\mathbb{F}_{q})\end{matrix}$

so that $K=I_{(n)}$ and $I=I_{(1^{n})}$. The double coset decompositions for
$\widetilde{G}\supseteq I$ and for $K\supseteq I$ are

$\widetilde{G}=\bigsqcup_{w\in\widetilde{W}}IwI\qquad\hbox{and}\qquad
K=\bigsqcup_{w\in W}IwI,$

where $\widetilde{W}$ is the affine Weyl group (the group of $n$-periodic
permutations, see [CR22, §5.1]) and $W$ is the finite Weyl group (the
symmetric group $S_{n}$).

Let $\mu$ be a partition of $n$. Let $u_{\mu}$ be a unipotent element in
$GL_{n}(\mathbb{F}_{q})$ in Jordan normal form with Jordan block sizes given
by the partition $\mu$. Since $GL_{n}(\mathbb{F}_{q})$ is a subgroup of
$\widetilde{G}=GL_{n}(\mathbb{F}_{q}((\epsilon))$ then $u_{\mu}$ is also an
element of $G$. The _$\pi$ -parabolic affine Springer fiber over $u_{\mu}$_ is

$Y_{I_{\pi}}^{-1}(u_{\mu})=\\{yI_{\pi}\in G/I_{\pi}\ |\ y^{-1}u_{\mu}y\in
I_{\pi}\\}.$ (5.1)

Let $w\in W$. The _affine Lusztig variety for $w$ and $u_{\mu}$_ is

$Y^{-1}_{IwI}(u_{\mu})=\\{yI\in G/I\ |\ y^{-1}u_{\mu}y\in IwI\\}.$ (5.2)

In [Me17, Theorem 5.15], Mellit gives precise meaning to
$\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))$ and shows that

$b_{\mu\pi}(t,q)=\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu})).$

We will not describe the weighted point count generating function that Mellit
uses for $\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))$ as it is a bit too
involved and would take us too far afield.

The point is that the affine Lusztig varieties $Y^{-1}_{IwI}(u_{\mu})$ for
$w\in W$ are related to the $\pi$-parabolic affine Springer fibers
$Y^{-1}_{I_{\pi}}(u_{\mu})$ considered in Mellit by expansion-contraction in
exactly the same way as the Lusztig varieties $Y^{-1}_{BwB}(u_{\mu})$ are
related to the $\pi$-parabolic Springer fibers $Y^{-1}_{P_{\pi}}(u_{\mu})$.
More precisely, affine versions of Proposition 2.3 and Corollary 2.2 hold
(with exactly the same proof) as follows: if $g\in\widetilde{G}$ then

$\mathrm{Card}(Y^{-1}_{I_{\pi}}(g))=\frac{1}{W_{\pi}(q)}\sum_{w\in
W_{\pi}}\mathrm{Card}(Y^{-1}_{IwI}(g)),$

and if $w\in W$ and $s_{i}$ is a simple reflection in $W$ such that
$\ell(s_{i}w)=\ell(w)+1$ then

$\displaystyle\mathrm{Card}(Y^{-1}_{Is_{i}wI}(g))$
$\displaystyle=\mathrm{Card}(Y^{-1}_{Iws_{i}I}(g)),$ if
$\ell(s_{i}w)=\ell(ws_{i})$, and
$\displaystyle\mathrm{Card}(Y^{-1}_{Is_{i}ws_{i}I}(g))$
$\displaystyle=q\mathrm{Card}(Y^{-1}_{Is_{i}wI}(g))+(q-1)\mathrm{Card}(Y^{-1}_{IwI}(g)),$
if $\ell(s_{i}ws_{i})=\ell(w)+2$.

This means that we can use Theorem 4.1 and the computation (compare to the
proof of Theorem 4.1)

$\displaystyle\sum_{\nu\vdash n}\sum_{w\in
S_{n}}q^{n(\mu)}a_{\mu\nu}(t,q^{-1})$ $\displaystyle
q^{n-\ell(\nu)}\kappa_{\nu,w}C_{w,\pi}\frac{1}{[\pi]!}=b_{\mu\pi}(t,q)$
$\displaystyle=\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))=\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))\mathrm{Card}(I_{\pi}/P)\frac{1}{[\pi]!}$
$\displaystyle=\sum_{w\in
S_{n}}\mathrm{Card}(Y^{-1}_{IwI}(u_{\mu}))C_{w,\pi}\frac{1}{[\pi]!}$
$\displaystyle=\sum_{\nu\vdash n}\sum_{w\in
S_{n}}\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))\kappa_{\nu,w}C_{w,\pi}\frac{1}{[\pi]!}$

to conclude that

$\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))=q^{n(\mu)}a_{\mu\nu}(t,q^{-1})q^{n-\ell(\nu)},$
(5.3)

where $\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))$ is a weighted point
count of exactly the same form as that used by Mellit.

###  5.1 A concluding sticking point

In the framework of this paper, extending from the Hall-Littlewood case to the
Macdonald polynomial case is achieved, geometrically, by going to the “affine
case” i.e., replacing $GL_{n}(\mathbb{F}_{q})$ by the loop group
$GL_{n}(\mathbb{F}_{q}((\epsilon))$. In the afffine case, the Lusztig variety
$Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})$ and the parabolic Springer fiber
$Y^{-1}_{I_{\pi}}(u_{\mu})$ are infinite, necessitating a more refined
definition of $\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))$ and
$\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))$. An attractive proposal is to
define these cardinalities as length generating functions over the affine Weyl
group,

$\displaystyle\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))$
$\displaystyle=\sum_{v\in\widetilde{W}}\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})\cap
IvI)\,t^{\ell(v)},$ $\displaystyle\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))$
$\displaystyle=\sum_{v\in\widetilde{W}}\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu})\cap
IvI)\,t^{\ell(v)}.$ (5.4)

This is not quite the same as the generating series used for the type $GL_{n}$
case by Mellit [Me17]. Mellit’s generating functions use $t^{\deg(v)}$, where
$\deg(v)$ is the degree of the corresponding vector bundle. Although Hikita
does not use vector bundles (and uses a Springer fiber over a semsimple
element instead of a nilpotent element), the grading parameter used by Hikita
[Hik12, Cor. 4.6 and Theorem 4.15] is similar to that used by Mellit [Me17].

Although the definition of $\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))$
using $t^{\ell(v)}$ is irresistible, it might be that a more correct way to
proceed is to use a statistic that connects to the representation theory of
the double affine Hecke algebra and define

$\displaystyle\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))$
$\displaystyle=\sum_{v\in\widetilde{W}}\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})\cap
IvI)\,t^{\mathrm{ord}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})\cap IvI)},$
$\displaystyle\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))$
$\displaystyle=\sum_{v\in\widetilde{W}}\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu})\cap
IvI)\,t^{\mathrm{ord}(Y^{-1}_{I_{\pi}}(u_{\mu})\cap IvI)},$ (5.5)

where $\mathrm{ord}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})\cap IvI)$ is the maximal
$k\in\mathbb{Z}_{\geq 0}$ such that

if $xvI\in Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})\cap IvI$ then
$x-1\in\epsilon^{-k}\mathrm{Lie}(K)$.

Here $K=G(\mathbb{F}_{q}[[\epsilon]])$, $\mathrm{Lie}(K)$ is the Lie algebra
of $K$ and $xvI$ denotes choosing representatives of elements of $IvI$ with

$x\quad\hbox{an element
of}\quad\prod_{\beta\in\mathrm{Inv}(v)}\mathcal{X}_{\beta},$

where $\mathcal{X}_{\beta}$ denotes the root subgroup corresponding to a root
$\beta$ and $\mathrm{Inv}(v)$ is the set of positive roots taken to negative
by $v$ (the ‘inversion set’ of $v$). This approach is attempting to capture a
statistic along the lines of what appears in [GMV14, Lemma 8.9], [OY14, Def.
8.1.1] and [VV07, second sentence of paragraph containing (2.4.6)]. Though not
exactly the same, the statistics $t^{\ell(v)}$ and
$t^{\mathrm{ord}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu})\cap IvI)}$ are closely
related because $\ell(t_{\lambda})=\langle\lambda,2\rho\rangle$ for a
translation $t_{\lambda}$ in the affine Weyl group (see [Mac03, (2,4,1)] and
[CR22, (5.3)]).

With these suggestions for $\mathrm{Card}(Y^{-1}_{I\gamma_{\nu}I}(u_{\mu}))$
and $\mathrm{Card}(Y^{-1}_{I_{\pi}}(u_{\mu}))$, the elements $A_{\mu}$ and
$M_{\mu}$ in the Hecke algebra which are defined in Section 1.3 will be, up to
a normalization by the cardinality of the centralizer of $u_{\mu}$, analogues
of integral form and modified form Macdonald polynomials for general Lie
types. The normalization by the centralizer of $u_{\mu}$ is a version of the
normalization by the constant $b_{\lambda}$ which appears in Macdonald’s book
(see [Mac, Ch. III (2.7), Ch. IV (2.6) and Ch. IV (7.3’)]).

In conclusion, I’d like to apologize, and thank my readers for their patience
with the wishywashyness and imprecision of this subsection. I had begun a more
in depth and careful treatment of some features but it quickly caused this
paper to balloon to an unpleasant and uncontrollable size. If someday I get
better at computing the combinatorics of affine Lusztig varieties efficiently
I will endeavor to tighten up these formulas. Even better would be if someone
who understands these objects better than me explains the picture properly. It
should be possible to derive Macdonald’s semistandard Young tableau formula
for the monomial expansion of the type $GL_{n}$ Macdonald polynomial by
counting points in affine Lusztig varieties (as was done for the nonaffine
case in [HR99, Theorem 3.4]).

##  6 Examples

###  6.1 Examples for $n=2$ and $n=3$

Integral Macdonald polynomial expansions in big Schurs: $J_{(1)}=S_{(1)}$

$\begin{array}[]{l}J_{(2)}=S_{(2)}+qS_{(1^{2})},\\\
J_{(1^{2})}=tS_{(2)}+S_{(1^{2})},\end{array}\qquad\qquad(K_{\lambda\mu}(q,t))=\begin{array}[]{c|cc}\lambda\backslash\mu&(2)&(1^{2})\\\
\hline\cr(2)&1&t\\\ (1^{2})&q&1\end{array}$
$\begin{array}[]{l}J_{(3)}=S_{(3)}+(q^{2}+q)S_{(21)}+q^{3}S_{(1^{3})},\\\
J_{(21)}=tS_{(3)}+(1+qt)S_{(21)}+qS_{(1^{3})},\\\
J_{(1^{3})}=t^{3}S_{(3)}+(t^{2}+t)S_{(21)}+S_{(1^{3})},\end{array}\qquad(K_{\lambda\mu}(q,t))=\begin{array}[]{c|ccc}\lambda\backslash\mu&(3)&(21)&(1^{3})\\\
\hline\cr(3)&1&t&t^{3}\\\ (21)&q+q^{2}&1+qt&t+t^{2}\\\
(1^{3})&q^{3}&q&1\end{array}$

Big Schur expansions in monomial symmetric functions: $S_{(1)}=(1-t)m_{(1)}$,

$\begin{array}[]{l}S_{(2)}=(1-t)m_{(2)}+(1-t)^{2}m_{(1^{2})},\\\
S_{(1^{2})}=(-t)(1-t)m_{(2)}+(1-t)^{2}m_{(1^{2})},\end{array}\qquad\qquad(L_{\lambda\nu}(t))=\begin{array}[]{c|cc}\lambda\backslash\nu&(2)&(1^{1})\\\
\hline\cr(2)&1&1\\\ (1^{2})&-t&1\end{array}$
$\begin{array}[]{l}S_{(3)}=(1-t)m_{(3)}+(1-t)^{2}m_{(21)}+(1-t)^{3}m_{(1^{3})},\\\
S_{(21)}=(-t)(1-t)m_{(3)}+(1-t)(1-t)^{2}m_{(21)}+2(1-t)^{3}m_{(1^{3})},\\\
S_{(1^{3})}=(-t)^{2}(1-t)m_{(3)}+(-t)(1-t)^{2}m_{(21)}+(1-t)^{3}m_{(1^{3})},\end{array}$
$(L_{\lambda\nu}(t))=\begin{array}[]{c|ccc}\lambda\backslash\nu&(3)&(21)&(1^{3})\\\
\hline\cr(3)&1&1&1\\\ (21)&0+(-t)&1+(-t)&1+1\\\
(1^{3})&(-t)^{2}&-t&1\end{array}$

Integral Macdonald polynomial expansions in monomial symmetric functions:
$J_{(1)}=m_{(1)}$,

$\begin{array}[]{l}J_{(2)}=(1-qt)(1-t)m_{2}+(1+q)(1-t)^{2}m_{(1^{2})},\\\
J_{(1^{2})}=(1+t)(1-t)^{2}m_{(1^{2})},\end{array}\qquad(a_{\mu\nu}(q,t))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&1-qt&1+q\\\ (1^{2})&0&1+t\end{array}$ $\displaystyle J_{(3)}$
$\displaystyle=(1-qt-q^{2}t+q^{3}t^{2})(1-t)m_{(3)}+(1+q-qt+q^{2}-q^{2}t-q^{3}t)(1-t)^{2}m_{(21)}$
$\displaystyle\qquad+(1+q)(1+q+q^{2})(1-t)^{3}m_{(1^{3})},$ $\displaystyle
J_{(21)}$
$\displaystyle=(1-qt^{2})(1-t)^{2}m_{(21)}+(2+q+t+2qt)(1-t)^{3}m_{(1^{3})},$
$\displaystyle J_{(1^{3})}$
$\displaystyle=(1+t)(1+t+t^{2})(1-t)^{3}m_{(1^{3})},$
$(a_{\mu\nu}(q,t))=\begin{array}[]{c|ccc}\mu\backslash\nu&(3)&(21)&(1^{3})\\\
\hline\cr(3)&(1-qt)(1-q^{2}t)&(1-qt)(1+q+q^{2})&(1+q)(1+q+q^{2})\\\
(21)&0&1-qt^{2}&2+t+q+2qt\\\ (1^{3})&0&0&(1+t)(1+t+t^{2})\end{array}$

Schur expansions in monomial symmetric functions: $s_{(1)}=m_{(1)}$,

$\begin{array}[]{l}s_{(2)}=m_{(2)}+m_{(1^{2})},\\\
s_{(1^{2})}=m_{(1^{2})},\end{array}\qquad\qquad(K_{\lambda\mu}(0,1))=\begin{array}[]{c|cc}\lambda\backslash\mu&(2)&(1^{2})\\\
\hline\cr(2)&1&1\\\ (1^{2})&0&1\end{array}$
$\begin{array}[]{l}s_{(3)}=m_{(3)}+m_{(21)}+m_{(1^{3})},\\\
s_{(21)}=m_{(21)}+2m_{(1^{3})},\\\
s_{(1^{3})}=m_{(1^{3})},\end{array}\qquad\qquad(K_{\lambda\mu}(0,1))=\begin{array}[]{c|ccc}\mu\backslash\lambda&(3)&(21)&(1^{3})\\\
\hline\cr(3)&1&1&1\\\ (21)&0&1&2\\\ (1^{3})&0&0&1\end{array}$

Modified Macdonald polynomial expansions in Schur functions:
$\widetilde{H}_{(1)}=s_{(1)}$,

$\begin{array}[]{l}\widetilde{H}_{(2)}=s_{(2)}+qs_{(1^{2})},\\\
\widetilde{H}_{(1^{2})}=s_{(2)}+ts_{(1^{2})},\\\
\end{array}\qquad\qquad(t^{n(\lambda)}K_{\lambda\mu}(q,t^{-1}))=\begin{array}[]{c|cc}\lambda\backslash\mu&(2)&(1^{2})\\\
\hline\cr(2)&1&1\\\ (1^{2})&q&t\end{array}$
$\begin{array}[]{l}\widetilde{H}_{(3)}=s_{(3)}+(q^{2}+q)s_{(21)}+q^{3}s_{(1^{3})},\\\
\widetilde{H}_{(21)}=s_{(3)}+(q+t)s_{(21)}+qts_{(1^{3})},\\\
\widetilde{H}_{(1^{3})}=s_{(3)}+(t^{2}+t)s_{(21)}+t^{3}s_{(1^{3})},\end{array}\qquad\qquad(t^{n(\lambda)}K_{\lambda\mu}(q,t^{-1}))=\begin{array}[]{c|ccc}\lambda\backslash\mu&(3)&(21)&(1^{2})\\\
\hline\cr(3)&1&1&1\\\ (21)&q^{2}+q&q+t&t^{2}+t\\\
(1^{3})&q^{3}&qt&t^{3}\end{array}$

Modified Macdonald polynomial expansions in monomial symmetric functions:
$\widetilde{H}_{(1)}=m_{(1)}$,

$\begin{array}[]{l}\widetilde{H}_{(2)}=m_{(2)}+(1+q)m_{(1^{2})},\\\
\widetilde{H}_{(1^{2})}=m_{(2)}+(1+t)m_{(1^{2})},\end{array}\qquad\qquad
b=(b_{\mu\nu}(q,t))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&1&1+q\\\ (1^{2})&1&1+t\end{array}$
$\begin{array}[]{l}\widetilde{H}_{(3)}=m_{(3)}+(1+q+q^{2})m_{(21)}+(1+q)(1+q+q^{2})m_{(1^{3})},\\\
\widetilde{H}_{(21)}=m_{(3)}+(1+q+t)m_{(21)}+(1+2(q+t)+qt)m_{(1^{3})},\\\
\widetilde{H}_{(1^{3})}=m_{(3)}+(1+t+t^{2})m_{(21)}+(1+t)(1+t+t^{2})m_{(1^{3})},\end{array}$
$b=(b_{\mu\nu}(q,t))=\begin{array}[]{c|ccc}\mu\backslash\nu&(3)&(21)&(1^{3})\\\
\hline\cr(3)&1&1+q+q^{2}&1+2(q^{2}+q)+q^{3}\\\ (21)&1&1+q+t&1+2(q+t)+qt\\\
(1^{3})&1&1+t+t^{2}&1+2(t+t^{2})+t^{3}\end{array}$

###  6.2 Hecke algebra elements and expansion-contraction

The irreducible characters of $H$ and the unipotent irreducible characters of
$G$ are given by

$\chi^{\lambda}_{H}(T_{\gamma_{\nu}})=L_{\lambda\nu}(q^{-1})q^{n-\ell(\nu)}\qquad\hbox{and}\qquad\chi^{\lambda}_{G}(u_{\mu})=q^{n(\mu)}K_{\lambda\mu}(0,q^{-1})$

and

$A_{\mu\nu}=\mathrm{Card}(Y^{-1}_{B\gamma_{\nu}B}(u^{-1}_{\mu}))=(q^{n(\mu)}a_{\mu\nu}(0,q^{-1})q^{n-\ell(\nu)}).$

Since

$A_{\mu\nu}=(\mathrm{tr}(u_{\mu}T_{\gamma_{\nu}},\mathbf{1}_{B}^{G}))=\sum_{\lambda}\chi^{\lambda}_{G}(u_{\mu})\chi^{\lambda}_{H}(T_{\gamma_{\nu}})\qquad\hbox{then}\qquad
A=\chi_{G}^{t}\chi_{H}.$

The Geck-Rouquier basis elements $\kappa_{\nu}$, the minimal central
idempotents $z_{\lambda}^{H}$, and the central elements $A_{\mu}$ are

$\kappa_{\nu}=\sum_{w}\kappa_{\nu,w}q^{-\ell(w)}T_{w^{-1}},\qquad
z^{H}_{\lambda}=\frac{\chi^{\lambda}_{G}(1)}{|G/B|}\sum_{\nu}\chi^{\lambda}_{H}(T_{\gamma_{\nu}})\kappa_{\nu}\qquad\hbox{and}\qquad
A_{\mu}=\sum_{\nu}A_{\mu\nu}\kappa_{\nu}.$

####  6.2.1 Type $GL_{2}(\mathbb{F}_{q})$

The parabolic projectors and the contraction matrix are

$\begin{array}[]{l}\mathbf{1}_{(1^{2})}=1,\\\
\mathbf{1}_{(2)}=1+T_{s_{1}},\end{array}\qquad\hbox{and}\qquad(C_{w,\pi})_{w\in
W,\pi\in\mathcal{P}}=\begin{array}[]{c|cccc}w\backslash\pi&(2)&(1^{2})\\\
\hline\cr s_{1}&1&0\\\ 1&1&1\end{array}$

The Geck-Rouquier elements and the expansion matrix are

$\begin{array}[]{l}\kappa_{s_{1}}=1\cdot q^{-1}T_{s_{1}}+0\cdot T_{1}\\\
\kappa_{1}=0\cdot
q^{-1}T_{s_{1}}+T_{1},\end{array}\qquad\kappa=(\kappa_{\nu,w})=\begin{array}[]{c|cc}\nu\backslash
w&s_{1}&1\\\ \hline\cr s_{1}&1&0\\\ 1&0&1\end{array}$

The minimal central idempotents in $H$ and the character table of the Hecke
algebra are

$\begin{array}[]{l}z^{H}_{(2)}=\frac{1}{1+q}(1+T_{s_{1}})=\frac{1}{1+q}(q\kappa_{s_{1}}+\kappa_{1}),\\\
z^{H}_{(1^{2})}=\frac{1}{1+q}(q-T_{s_{1}})=\frac{q}{1+q}(-\kappa_{s_{1}}+\kappa_{1})\end{array}\qquad\quad\chi_{H}=(\chi^{\lambda}_{H}(T_{\gamma_{\nu}}))=\begin{array}[]{c|cc}\lambda\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&q&1\\\ (1^{2})&-1&1\end{array}$

The unipotent character table of $G$ is

$\chi_{G}=(\chi^{\lambda}_{G}(u_{\mu}))=\begin{array}[]{c|cc}\lambda\backslash\mu&(2)&(1^{2})\\\
\hline\cr(2)&1&1\\\ (1^{2})&0&q\end{array}$

The central elements $A_{\mu}$ are

$\begin{array}[]{l}A_{(2)}=q\kappa_{s_{1}}+\kappa_{1}=T_{s_{1}}+1,\\\
A_{(1^{2})}=(1+q)\kappa_{1}=[2]T_{1},\end{array}\qquad(A_{\mu\nu}(q))=(\mathrm{tr}(u_{\mu}T_{\gamma_{\nu}},\mathbf{1}_{B}^{G}))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&q&1\\\ (1^{2})&0&q+1\end{array}\qquad\qquad$

####  6.2.2 Type $GL_{3}(\mathbb{F}_{q})$

The parabolic projectors and the contraction matrix are

$\begin{array}[]{l}\mathbf{1}_{(3)}=T_{s_{1}s_{2}}+T_{s_{2}s_{1}}+T_{s_{1}s_{2}s_{1}}+T_{s_{1}}+T_{s_{2}}+T_{1},\\\
\mathbf{1}_{(21)}=T_{s_{1}}+T_{1},\\\
\mathbf{1}_{(1^{3})}=T_{1},\end{array}\quad\hbox{and}\quad(C_{w,\pi})=\begin{array}[]{c|ccc}w\backslash\pi&(3)&(21)&(1^{3})\\\
\hline\cr s_{1}s_{2}&1&0&0\\\ s_{2}s_{1}&1&0&0\\\ s_{1}s_{2}s_{1}&1&0&0\\\
s_{2}&1&0&0\\\ s_{1}&1&1&0\\\ 1&1&1&1\end{array}$

The Geck-Rouquier elements and the expansion matrix are

$\begin{array}[]{l}\kappa_{s_{1}s_{2}}=1\cdot q^{-2}T_{s_{1}s_{2}}+1\cdot
q^{-2}T_{s_{2}s_{1}}+(q-1)q^{-3}T_{s_{1}s_{2}s_{1}},\\\ \kappa_{s_{1}}=q\cdot
q^{-3}T_{s_{1}s_{2}s_{1}}+1\cdot q^{-1}T_{s_{1}}+1\cdot q^{-1}T_{s_{2}},\\\
\kappa_{1}=T_{1},\end{array}\qquad\begin{array}[]{c|cccccc}\nu\backslash
w&s_{1}s_{2}&s_{2}s_{1}&s_{1}s_{2}s_{1}&s_{2}&s_{1}&1\\\ \hline\cr
s_{1}s_{2}&1&1&q-1&0&0&0\\\ s_{1}&0&0&q&1&1&0\\\ 1&0&0&0&0&0&1\end{array}$

The minimal central idempotents and the character table of the Hecke algebra
are

$\displaystyle\begin{array}[]{l}z^{H}_{(1^{3})}=\frac{1}{[3]!}(q^{2}\kappa_{s_{1}s_{2}}+q\kappa_{s_{1}}+\kappa_{1}),\\\
z^{H}_{(21)}=\frac{q[2]}{[3]!}(-q\kappa_{s_{1}s_{2}}+(q-1)\kappa_{s_{1}}+2\kappa_{1}),\\\
z^{H}_{(3)}=\frac{q^{3}}{[3]!}(\kappa_{s_{1}s_{2}}-\kappa_{s_{1}}+\kappa_{1}),\end{array}\qquad\chi_{H}=(\chi^{\lambda}_{H}(T_{\gamma_{\nu}}))=\begin{array}[]{c|ccc}\lambda\backslash\nu&(3)&(21)&(1^{3})\\\
\hline\cr(3)&q^{2}&q&1\\\ (21)&-q&q-1&2\\\ (1^{3})&1&-1&1\end{array}$

The unipotent character table of $G$ is

$\chi_{G}=(\chi^{\lambda}_{G}(u_{\mu}))=\begin{array}[]{c|ccc}\lambda\backslash\mu&(3)&(21)&(1^{3})\\\
\hline\cr(3)&1&1&1\\\ (21)&0&q&q^{2}+q\\\ (1^{3})&0&0&q^{3}\end{array}$

The central elements $A_{\mu}\in Z(H)$ are

$\displaystyle A_{(3)}$
$\displaystyle=T_{s_{1}s_{2}}+T_{s_{2}s_{1}}+T_{s_{1}s_{2}s_{1}}+T_{s_{1}}+T_{s_{2}}+T_{1}=q^{2}\kappa_{s_{1}s_{2}}+q\kappa_{s_{1}}+\kappa_{1},$
$\displaystyle A_{(21)}$
$\displaystyle=T_{s_{1}s_{2}s_{1}}+qT_{s_{1}}+qT_{s_{2}}+(1+2q)T_{1}=q^{2}\kappa_{s_{1}}+(1+2q)\kappa_{1},$
$\displaystyle A_{(1^{3})}$ $\displaystyle=[3]!T_{1}=[3]!\kappa_{1},$

and the table of bitraces is

$A=(A_{\mu\nu}(q))=(\mathrm{tr}(u_{\mu}T_{\gamma_{\nu}},\mathbf{1}_{B}^{G})=(q^{n(\mu)+n-\ell(\nu)}a_{\mu\nu}(0,q^{-1}))=\begin{array}[]{c|ccc}\mu\backslash\nu&(3)&(21)&(1^{3})\\\
\hline\cr(3)&q^{2}&q&1\\\ (21)&0&q^{2}&2q+1\\\ (1^{3})&0&0&[3][2]\end{array}$

Let us verify that

$A=(A_{\mu\nu})=\chi_{G}^{t}\chi_{H}=\begin{pmatrix}1&0&0\\\ 1&q&0\\\
1&q^{2}+q&q^{3}\end{pmatrix}\begin{pmatrix}q^{2}&q&1\\\ -q&q-1&2\\\
1&-1&1\end{pmatrix}=\begin{pmatrix}q^{2}&q&1\\\ 0&q^{2}&2q+1\\\
0&0&[3][2]\end{pmatrix}\quad\hbox{and}$
$\chi_{H}=L_{\lambda\nu}(q^{-1})\mathrm{diag}(q^{n-\ell(\nu)})=\begin{pmatrix}1&1&1\\\
-q^{-1}&1-q^{-1}&2\\\ q^{-2}&-q&1\end{pmatrix}\begin{pmatrix}q^{2}&0&0\\\
0&q&0\\\ 0&0&1\end{pmatrix}=\begin{pmatrix}q^{2}&q&1\\\ -q&q-1&2\\\
1&-1&1\end{pmatrix}.$

Multiplying $A$ by the matrix $\kappa=(\kappa_{\nu,w})$ gives

$(A_{\mu
w}(q))=(\\#Y^{-1}_{BwB}(u_{\mu}))=(\mathrm{tr}(u_{\mu}T_{w},\mathbf{1}_{B}^{G})=\begin{array}[]{c|cccccc}\mu\backslash
w&s_{1}s_{2}&s_{2}s_{1}&s_{1}s_{2}s_{1}&s_{2}&s_{1}&1\\\
\hline\cr(3)&q^{2}&q^{2}&q^{3}&q&q&1\\\ (21)&0&0&q^{3}&q^{2}&q^{2}&2q+1\\\
(1^{3})&0&0&0&0&0&[3][2]\end{array}$

since $(q-1)\cdot q^{2}+q\cdot q=q^{3}$, $(q-1)\cdot 0+q\cdot q^{2}=q^{3}$ and
$(q-1)\cdot 0+q\cdot 0=0$. Note that the row sums of this matrix are all
$[3][2]$. Then, multiplying by the contraction matrix gives

$(\\#Y^{-1}_{P_{\pi}}(u_{\mu}))=\begin{array}[]{c|cccc}\mu\backslash\pi&Y^{-1}_{(1^{3})}&Y^{-1}_{(21)}&Y^{-1}_{(12)}&Y^{-1}_{(3)}\\\
\hline\cr(3)&1&1+q&1+q&[3][2]\\\ (21)&2q+1&q^{2}+2q+1&q^{2}+2q+1&[3][2]\\\
(1^{3})&[3][2]&[3][2]&[3][2]&[3][2]\end{array}$

####  6.2.3 Check that $M\kappa CDFL=1$ for $n=2$ and $n=3$

The expansion and contraction matrices for $n=2$ are

$\kappa=(\kappa_{\nu,w})_{\nu\in\mathcal{W},w\in
W}=\begin{array}[]{c|cc}\nu\backslash w&s_{1}&1\\\ \hline\cr s_{1}&1&0\\\
1&0&1\end{array}\qquad\hbox{and}\qquad(C_{w,\pi})_{w\in
W,\pi\in\mathcal{P}}=\begin{array}[]{c|cccc}w\backslash\pi&(2)&(1^{2})\\\
\hline\cr s_{1}&1&0\\\ 1&1&1\end{array}$

Using notations as in the proof of Theorem 4.1, the product

$M\kappa CD=\begin{pmatrix}q&0\\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{pmatrix}1&0\\\
1&1\end{pmatrix}\begin{pmatrix}\frac{1}{[2]}&0\\\
0&1\end{pmatrix}=\begin{pmatrix}\frac{q}{[2]}&0\\\
\frac{1}{[2]}&1\end{pmatrix}$

is the inverse of

$FL=\begin{pmatrix}1&-1\\\ 0&1\end{pmatrix}\begin{pmatrix}1&1\\\
-q^{-1}&1\end{pmatrix}=\begin{pmatrix}1+q^{-1}&0\\\ -q^{-1}&1\end{pmatrix}.$

From Section 6.1,

$(a_{\mu\nu}(q,t))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&1-qt&1+q\\\ (1^{2})&0&1+t\end{array}\qquad\hbox{so
that}\qquad(a_{\mu\nu}(t,q^{-1}))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&1-tq^{-1}&1+t\\\ (1^{2})&0&1+q^{-1}\end{array}$

and

$(q^{n(\mu)}a_{\mu\nu}(t,q^{-1}))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&1-tq^{-1}&1+t\\\ (1^{2})&0&1+q\end{array}$

and

$(q^{n(\mu)}a_{\mu\nu}(t,q^{-1})q^{n-\ell(\nu)})=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&q(1-tq^{-1})&1+t\\\ (1^{2})&0&1+q\end{array}$

From Section 6.1,

$b=(b_{\mu\nu}(q,t))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&1&1+q\\\ (1^{2})&1&1+t\end{array}\qquad\hbox{so
that}\qquad(b_{\mu\nu}(t,q))=\begin{array}[]{c|cc}\mu\backslash\nu&(2)&(1^{2})\\\
\hline\cr(2)&1&1+t\\\ (1^{2})&1&1+q\end{array}$

The expansion matrix and contraction matrices for $n=3$ are

$(\kappa_{\nu,w})=\begin{array}[]{c|cccccc}\nu\backslash
w&s_{1}s_{2}&s_{2}s_{1}&s_{1}s_{2}s_{1}&s_{2}&s_{1}&1\\\ \hline\cr
s_{1}s_{2}&1&1&q-1&0&0&0\\\ s_{1}&0&0&q&1&1&0\\\
1&0&0&0&0&0&1\end{array}\quad\hbox{and}\quad(C_{w,\pi})=\begin{array}[]{c|ccc}w\backslash\pi&(3)&(21)&(1^{3})\\\
\hline\cr s_{1}s_{2}&1&0&0\\\ s_{2}s_{1}&1&0&0\\\ s_{1}s_{2}s_{1}&1&0&0\\\
s_{2}&1&0&0\\\ s_{1}&1&1&0\\\ 1&1&1&1\end{array}$

Using notations as in the proof of Theorem 4.1, the product

$\displaystyle M\kappa CD$ $\displaystyle=\begin{pmatrix}q^{2}&0&0\\\ 0&q&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&1&q-1&0&0&0\\\ 0&0&q&1&1&0\\\
0&0&0&0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\\ 1&0&0\\\ 1&0&0\\\ 1&0&0\\\
1&1&0\\\ 1&1&1\end{pmatrix}\begin{pmatrix}\frac{1}{[3][2]}&0&0\\\
0&\frac{1}{[2]}&0\\\ 0&0&1\end{pmatrix}$
$\displaystyle=\begin{pmatrix}q^{2}&0&0\\\ 0&q&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}q+1&0&0\\\ q+2&1&0\\\
1&1&1\end{pmatrix}\begin{pmatrix}\frac{1}{[3][2]}&0&0\\\ 0&\frac{1}{[2]}&0\\\
0&0&1\end{pmatrix}=\begin{pmatrix}\frac{q^{2}}{[3]}&0&0\\\
\frac{q^{2}+2q}{[3][2]}&\frac{q}{[2]}&0\\\
\frac{1}{[3][2]}&\frac{1}{[2]}&1\end{pmatrix}$

is the inverse of

$FL=\begin{pmatrix}1&-1&1\\\ 0&1&-2\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&1&1\\\ -q^{-1}&1-q^{-1}&2\\\
q^{-2}&-q^{-1}&1\end{pmatrix}=\begin{pmatrix}q^{-2}[3]&0&0\\\
-q^{-2}(q+2)&q^{-1}(q+1)&0\\\ q^{-2}&-q^{-1}&1\end{pmatrix}.$

## References

  * [Bou] N. Bourbaki, Groupes et algèbres de Lie, vol. 4–6, Masson 1981, MR0647314
  * [CR22] L. Colmenarejo and A. Ram, _$c$ -functions and Macdonald polynomials_, arxiv:2212.03312.
  * [CR81] C.W. Curtis and I. Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orders, Wiley Classics Lib. I and II, John Wiley & Sons, New York, 1981. MR0892316.
  * [DL76] P. Deligne and G. Lusztig, _Representations of reductive groups over finite fields_ , Ann. of Math. (2) 103 (1976) 103–161, MR0393266.
  * [DRS22] P. Diaconis, A. Ram and M. Simper, _Double coset Markov chains_ , Forum Math. Sigma 11 (2023), Paper No. e2, 45 pp., MR4530094, arXiv2208.10699.
  * [GH93] A. Garsia and M. Haiman, _A graded representation model for Macdonald’s polynomials_ , Proc. Nat. Acad. Sci. USA 90 (1993) 3607-3610. MR1214091
  * [GH96] A. Garsia and M. Haiman, _A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion_, J. Algebraic Combinatorics 5 (1996) 191-244, MR1394305.
  * [GR05] A. Garsia and J.B. Remmel, _Breakthroughs in the theory of Macdonald polynomials_ , Proc. Nat. Acad. Sci. USA 102 (2005) 3891-3894, MR2139721.
  * [GP00] M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monogr. (N.S.) 21 The Clarendon Press, Oxford University Press, New York, 2000, xvi+446 pp. ISBN: 0-19-850250-8, MR1778802.
  * [GMV14] E. Gorsky, M. Mazin and M. Vazirani, _Affine permutations and rational slope parking functions_ , Trans. Amer. Math. Soc. 368 (2016) 8403-8445, MR3551576, arXiv:140303.
  * [HR99] T. Halverson and A. Ram, _Bitraces for $GL_{n}(\mathbb{F}_{q})$ and the Iwahori-Hecke algebra of type $A_{n-1}$_, Indag. Mathem. N.S. 10 (1999) 247–268, MR1816219.
  * [HR04] T. Halverson and A. Ram, _Partition algebras_ , European J. of Combinatorics, 26(2005) 869-921, MR2143201, arxic:0401314.
  * [HLR] T. Halverson, R. Leduc and A. Ram, _Iwahori-Hecke algebras of type A, bitraces and symmetric functions_ , Int. Math. Research Notices (1997) 401–416. MR1443319.
  * [He23] X. He, _On affine Lusztig varieties_ , arXiv:2302.03203.
  * [Hik12] T. Hikita, _Affine Springer fibers of type $A$ and combinatorics of diagonal coinvariants_, Adv. Math. 263 (2014) 88–122, MR3239135, arXiv:1203.5878.
  * [HS79] R. Hotta, N. Shimomura, _The fixed point subvarieties of unipotent transformations on generalized flag varieties and the Green functions_ , Math. Ann. 241 (1979) 193-208.
  * [LLT95] A. Lascoux, B. Leclerc and J.-Y. Thibon, _Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties_ , J. Math. Phys. 38 (1997) 1041–1068. MR1434225, arXiv:q-alg/9512031.
  * [Lu81] G. Lusztig, _Green polynomials and singularities of unipotent classes_ , Adv. Math. 42 (1981) 169-178, MR0641425.
  * [Lu21] G. Lusztig, _Traces on Iwahori-Hecke algebras and counting rational points_ , arXiv:2105.04061.
  * [Mac] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford Mathematical Monographs, Oxford University Press, New York, 1995. ISBN: 0-19-853489-2, MR1354144.
  * [Mac03] I.G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003. MR1976581.
  * [Me17] A. Mellit, _Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers_ , Ann. of Math. (2) 192 (2020), 165-228. MR4125451, arxiv:1710.04513.
  * [OY14] A. Oblomkov and Z. Yun, _Geometric representations of graded and rational Cherednik algebras_ , Adv. Math. 292 (2016) 601–706, MR3464031, arxiv:1407.5685.
  * [Ra91] A. Ram, _A Frobenius formula for the characters of the Hecke algebras_ , Invent. Math. 106 (1991), 461-488, MR1134480.
  * [VV07] M. Varagnolo and E. Vasserot, _Finite dimensional representations of DAHA and affine Springer fibers: The spherical case_ , Duke Math. J. 147 (2009) 439-540, MR2510742, arxiv: 0705.2691.
  * [WW12] J. Wan and W. Wang, _Frobenius map for the centers of Hecke algebras_ , Trans. Amer. Math. Soc. 367 (2015) 5507?5520, MR3347181, arxiv:1208.4446.


# Affect in Tweets Using Experts Model

Subba Reddy Oota
IIIT-Hyderabad
Hyderabad, India
<EMAIL_ADDRESS>
&Adithya Avvaru
IIIT-Hyderabad
Hyderabad, India
<EMAIL_ADDRESS>
Mounika Marreddy
IIIT-Hyderabad
Hyderabad, India
<EMAIL_ADDRESS>
&Radhika Mamidi
IIIT-Hyderabad
Hyderabad, India
<EMAIL_ADDRESS>

###### Abstract

Estimating the intensity of emotion has gained significance as modern textual
inputs in potential applications like social media, e-retail markets,
psychology, advertisements etc., carry a lot of emotions, feelings,
expressions along with its meaning. However, the approaches of traditional
sentiment analysis primarily focuses on classifying the sentiment in general
(positive or negative) or at an aspect level(very positive, low negative,
etc.) and cannot exploit the intensity information. Moreover, automatically
identifying emotions like anger, fear, joy, sadness, disgust etc., from text
introduces challenging scenarios where single tweet may contain multiple
emotions with different intensities and some emotions may even co-occur in
some of the tweets. In this paper, we propose an architecture, Experts Model,
inspired from the standard Mixture of Experts (MoE) model. The key idea here
is each expert learns different sets of features from the feature vector which
helps in better emotion detection from the tweet. We compared the results of
our Experts Model with both baseline results and top five performers of
SemEval-2018 Task-1, Affect in Tweets (AIT). The experimental results show
that our proposed approach deals with the emotion detection problem and stands
at top-5 results.

##  1 Introduction

Sentiment analysis is one of the most famous Natural Language Processing (NLP)
tasks. This task was used in social network services [Pang et al., 2008, Asur
and Huberman, 2010], e-retailing, advertising [Qiu et al., 2010, Jin et al.,
2007], question answering systems [Somasundaran et al., 2007, Stoyanov et al.,
2005] and many other domains. It focuses on the automatic prediction of
polarity or sentiment on tweets or reviews. While most computer science
research in this field has focused on strict positive/negative sentiment
analysis, the three dominant theories [Marsella et al., 2010, Stelmack and
Stalikas, 1991] of emotion agree that humans express or operate with much more
nuanced emotion representations. In other words, tweets or reviews, in recent
times, include non-standard representations of emotion like emoticons, emojis
etc. This task of sentiment analysis became increasingly complex due to an
addition of creatively spelt words (for eg, “gm” for “good morning”, “hpy” for
“happy” etc,.) and hashtags, particularly in case of tweets.

The current research of Sentiment Analysis is gearing towards evaluating
emotion intensity in a text to identify and quantify discrete emotions which
can help in above mentioned applications mentioned and many new ones. Here,
intensity refers to the degree or quantity of an emotion such as anger, fear,
joy, or sadness. For example, consider the three statements “The product is
awesome and delivery is before time”, “It was waste of money and time” and
“This TV is ok ok product at this budget range”. The above 3 statements
respectively express the level of satisfaction as very happy, very sad and
moderately happy. This illustrates the different intensities of happiness of
the particular person. Similarly, a person expresses different intensities of
other emotions like anger, frustration etc.

##  2 Related Work

In literature, there has been an increasing focus towards building sentiment
classification/prediction models through various approaches like rule mining,
machine learning or deep learning. A brief overview of the efforts of
scientific community towards sentiment related models can be found in [Pang et
al., 2008, Paltoglou et al., 2010, Wilson et al., 2004, Liu and Zhang, 2012].

Many prior works of emotion detection have always used manual strategies to
map emotion category to emotional expression. However, such manual
categorization requires an understanding of the emotional content of each
expression, which is time-consuming and an arduous task. In [Warriner et al.,
2013], emotions are projected as points in 3-dimensional space of valence
(positiveness-negativeness), arousal (active-passive), and dominance
(dominant-submissive). Using this theory, there is a huge effort on creating
valence lexicons like MPQA [Wilson et al., 2005], Norms Lexicon [Warriner et
al., 2013], NRC Emotion Lexicon [Mohammad and Bravo-Marquez, 2017a], WordNet
Affect Lexicon [Baccianella et al., 2010] and many others. However, these
lexicon based approaches usually ignore the intensity of emotions and
sentiment, which provides important information for fine-grained sentiment
analysis. The current research shifts towards automatic emotion classification
which has been proposed for many different kinds of text, including tweets
[Mohammad and Kiritchenko, 2015, Mohammad and Bravo-Marquez, 2017a].

Existing approaches to analyze intensity are based simply on lexicons, word-
embeddings, combinational features and supervised learning. [Nielsen, 2011]
introduced lexicon based methods which rely on lexicons to assign the
intensity score of each word in the tweet. However, this method did not
consider the semantic information from the text. Some supervised methods like
deep neural networks were applied to tweet sentiment analysis to predict the
polarity [dos Santos and Gatti, 2014]. Although deep learning methods
outperform lexicon based methods as shown in [dos Santos and Gatti, 2014], but
could not capture the fine-grained property of the sentiment in a text. To
capture this fine-grained aspect of a sentiment, [Mohammad, 2016] proposed to
identify the intensity of emotion in texts. To further expand the scope of
emotion analysis, [Mohammad and Bravo-Marquez, 2017b, Mohammad et al., 2018]
introduced EmoInt-2017 and SemEval-2018 shared tasks where the top performing
teams use deep learning models such as CNN, RNN, LSTMs [Goel et al., 2017,
Köper et al., 2017] and classifiers like Support Vector Machine or Random
Forest [Duppada and Hiray, 2017, Köper et al., 2017]. In the above two tasks,
some participants use an ensemble-based approach by simply averaging the
outputs of two top performing models [Duppada and Hiray, 2017, Duppada et al.,
2018] and the weighted average of predicted outputs of three different deep
neural network based models [Goel et al., 2017]. The subtasks of SemEval-2018
Task-1, AIT [Mohammad et al., 2018] are detailed in Section 3.

The structure of the paper is as follows. In section 3, we describe the
dataset. Section 4 describes the approach we are using to build the model,
while section 5 discusses the approaches of preprocessing and feature
extraction. Section 6 presents comparative results of various models along
with the analysis of the results. Section 7 presents concluding remarks and
future work.

##  3 Dataset Description

We used the dataset from SemEval-2018 Task 1: AIT
111https://competitions.codalab.org/competitions/17751#learn_the_details-
datasets for training our system. There is a total of five subtasks: EI-reg
(Emotion Intensity regression), EI-oc (Emotion Intensity ordinal
classification), V-reg (Valence regression), V-oc (Valence ordinal
classification) and E-c (Emotion multi-label classification). Each subtask has
three datasets: train, dev, and test. In this paper, we worked on the all five
subtasks mentioned above. The dataset details are briefly shown in Table 1.

Dataset | Train | Dev | Test | Total
---|---|---|---|---
EI-reg, EI-oc |  |  |  |
anger | 1701 | 388 | 1002 | 3091
fear | 2252 | 389 | 986 | 3011
joy | 1616 | 290 | 1105 | 2905
sadness | 1533 | 397 | 975 | 2905
V-reg, V-oc | 1181 | 886 | 3259 | 2567
E-c | 6838 | 886 | 3259 | 10953
Table 1: SemEval-2018 Task-1 Dataset Details

##  4 Approach

We took inspiration from the Mixture of Experts (MoE) [Jacobs et al., 1991,
Nowlan and Hinton, 1991] regression and classification models, where each
expert tunes to some set of features out of all the features.

###  4.1 MoE Description

In this subsection, we briefly describe the MoE model to enable the readers to
relate our model to MoE architecture. The MoE architecture consists of a
number of experts and a gating network. In MoE, there are parameters for each
of the expert and a separate set of parameters for gating network. The expert
and gate parameters are trained simultaneously using Expectation Maximization
[Jordan and Jacobs, 1994] or Gradient Descent Approach [Jordan and Xu, 1995].

Consider the following regression problem. Let
$X=\\{\mathbf{x}^{(n)}\\}_{n=1}^{N}$ are $N$ input vectors (samples) and
$Y=\\{\mathbf{y}^{(n)}\\}_{n=1}^{N}$ are $N$ targets for each input vector.
Then, MoE model is described in terms of parameter
$\theta=\\{\theta_{g},\theta_{e}\\}$ where $\theta_{g}$ is set of the gate
parameters and $\theta_{e}$ is sets of the expert parameters. Given a sample
$\mathbf{x}$ from among $N$ samples, the total probability of predicting
target $\mathbf{y}$ can be written in terms of the experts as

$\displaystyle P(\mathbf{y}|\mathbf{x},\theta)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{I}P(\mathbf{y},\mathbf{x}|\theta)$ (1)
$\displaystyle=$
$\displaystyle\sum_{i=1}^{I}P(i|\mathbf{x},\theta_{g})P(\mathbf{y}|i,\mathbf{x},\theta_{e})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{I}g_{i}(\mathbf{x},\theta_{g})P(\mathbf{y}|i,\mathbf{x},\theta_{e})$

where $I$ is the number of experts, the function
$g_{i}(\mathbf{x},\theta_{g})=P(i|\mathbf{x},\theta_{g})$ represents the
probability of selecting $i^{th}$ expert given $x$ and
$P(\mathbf{y}|i,\mathbf{x},\theta_{e})$ represents the probability of $i^{th}$
expert giving $\mathbf{y}$ on seeing $\mathbf{x}$.

The MoE training maximizes the log-likelihood of the probability in equation 1
to learn the parameters $\theta$ [Yuksel et al., 2012].

Figure 1:  Proposed Experts model

###  4.2 Our Proposed Approach

We used the similar architecture, however with some modifications, to measure
the intensity of an emotion in a tweet (regression) or predict an emotional
intensity (classification). In our proposed approach, we pre-train each
expert, to get parameters $\theta_{e}$, on the training samples, unlike
traditional MoE model. Each expert, in itself, can be a separate
Regression/Classification model like Multi-level Perceptron (MLP) model or
Long Short-Term Memory (LSTM) model or any other model that best suits the
data and task at hand. Once each expert is trained separately, we train only
the gating network based on Gradient Descent Approach. The Detailed
description of the model is depicted in Figure 1 and explained below.

We build different models - Neural Network Classifier/Regressor, Gradient
Boosting Classifier/Regressor, XGBoost Classifier/Regressor, Random Forest
Classifier/Regressor, Lasso Regressor and Light Gradient Boosting
Classifier/Regressor and train each of them with the extracted feature vector
of each tweet. We obtain this feature vector by concatenating all of the
features discussed in Section 5. We assign parameters $\theta_{g}$, weights
and bias, for each Classifier/Regressor at the gating network. Later, we train
the gating network to fit the predicted $\hat{y}$ of each expert $i$ with
actual $y$ and learn the best $\theta_{g}$.

Let $w[i]$ denote weight of each expert $i$ at the gating network. Let $b[i]$
denote the bias term for each expert $i$ at the gating network. Let $I$ be the
number of experts. We define the Error function ($E$) as

$E=\sum_{i=1}^{I}\frac{1}{2}prob[i]\Big{(}\mathbf{y}[i]-\hat{\mathbf{y}}[i]+b[i]\Big{)}^{2}$

where $prob[i]$ is softmax probability of weight $w[i]$, $\mathbf{y}[i]$ is
the actual $\mathbf{y}$ of $i^{th}$ expert for some sample $\mathbf{x}$.
Similarly, $\hat{\mathbf{y}}[i]$ is the predicted $\mathbf{y}$ of $i^{th}$
expert for same sample $\mathbf{x}$. It is to be noted that $\forall i\hskip
3.0pt\mathbf{y}[i]=\mathbf{y}$.

For each sample $\mathbf{x}$ and $\mathbf{y}$, we train the gating network
using the update equations of gradients as follows:

$\displaystyle\frac{\partial E}{\partial w[i]}$
$\displaystyle=\frac{1}{2}prob[i]\Big{(}1-prob[i]\Big{)}\Big{(}\mathbf{y}[i]-\hat{\mathbf{y}}[i]+b[i]\Big{)}^{2}$
$\displaystyle\frac{\partial E}{\partial b[i]}$
$\displaystyle=prob[i]\Big{(}\mathbf{y}[i]-\hat{\mathbf{y}}[i]+b[i]\Big{)}$

and

$\displaystyle w[i]$ $\displaystyle=w[i]-\eta*\frac{\partial E}{\partial
w[i]}$ $\displaystyle b[i]$ $\displaystyle=b[i]-\eta*\frac{\partial
E}{\partial b[i]}$

where $\eta$ is the learning rate.

##  5 Preprocessing & Feature Extraction

To preprocess each tweet, first we break all the contractions (like “can’t” to
“cannot”, “I’m” to “I am” etc,.) followed by spelling corrections, decoding
special words and acronyms (like “e g” to “eg”, “fb” to “facebook” etc,.) and
symbol replacements (like “$” to “dollar”, “=” to “is equal to” etc). Later,
we tokenized each tweet using NLTK tweet tokenizer
222https://www.nltk.org/api/nltk.tokenize.html.

The basic idea of using different experts and eclectic features is from the
intuition that each expert learn from different aspects of the concatenated
features. Hence, we explored and extracted a variety of features; and used
only the features which best performs among all the explored ones are
explained in the following subsections.

###  5.1 Deep-Emoji Features

Deep-Emoji [Felbo et al., 2017] performs prediction using the model trained on
a dataset of 1246 million tweets and achieves state-of-the-art performance
within sentiment, emotion and sarcasm detection. We can use the architecture
of Deep-Emoji and train the model using millions of tweets from social media
to get a better representation of new data. Using the pre-trained Deep-Emoji
model, we extracted two different set of features - one, 64-dimensional vector
from the softmax layer and the other, 2304-dimensional vector from attention
layer.

###  5.2 Word-Embedding Features

In this paper, we tried four different pre-trained word-embedding approaches
such as Word2Vec [Mikolov et al., 2013], GloVe [Pennington et al., 2014],
Edinburgh Twitter Corpus [Petrović et al., 2010] and FastText [Bojanowski et
al., 2017] for generating word vectors. We used the GloVe model of 300
dimensions.

###  5.3 Skip-Thought Features

Skip-Thoughts vectors [Kiros et al., 2015] model is in the framework of
encoder-decoder models. Here, an encoder maps words to sentence vector and a
decoder is used to generate the surrounding sentences. The main advantage of
Skip-Thought vectors is that it can produce highly generic sentence
representations from an encoder that share both semantic and syntactic
properties of surrounding sentences. Here, we used Skip-Thought vector encoder
model to produce a 4800 dimension vector representation of each tweet.

###  5.4 Lexicon Features

We also chose various lexicon features for the model. The lexicon features
include AFINN Lexicon [Nielsen, 2011] (calculates positive and negative
sentiment scores from the lexicon), MPQA Lexicon [Wilson et al., 2005]
(calculates the number of positive and negative words from the lexicon), Bing
Liu Lexicon [Bauman et al., 2017] (calculates the number of positive and
negative words from the lexicon), NRC Affect Intensities, NRC-Word-Affect
Emotion Lexicon, NRC Hash-tag Sentiment Lexicon, Sentiment140 Lexicon [Go et
al., 2009] (calculates positive and negative sentiment score provided by the
lexicon in which tweets are annotated by lexicons), and SentiWordNet
[Baccianella et al., 2010] (calculates positive, negative, and neutral
sentiment score). The final feature vector is the concatenation of all the
individual features.

###  5.5 Hash-tag Intensity Features

The work by [Mohammad and Bravo-Marquez, 2017a] describes that removal of the
emotion word hashtags causes the emotional intensity of the tweet to drop.
This indicates that emotion word hashtags are not redundant with the rest of
the tweet in terms of the overall intensity. Here, we used Depeche mood
dictionary [Staiano and Guerini, 2014] to get the intensities of hashtag
words. We average the intensities of all hashtags of a single tweet to get the
total intensity score.

###  5.6 Stylometric Features

Tweets and other electronic messages (e-mails, posts, etc.) are written far
shorter, way more informal and much richer in terms of expressive elements
like emoticons and aspects at both syntax and structure level, etc. Common
techniques use stylometric features [Anchiêta et al., 2015] which are
categorized into 5 different types: lexical, syntactic, structural, content
specific, and idiosyncratic. In this paper, we used 7 stylometric features
such as “number of emoticons”, “number of nouns”, “number of adverbs”, “number
of adjectives”, “number of punctuations”, “number of words”, and “average word
length”.

###  5.7 Unsupervised Sentiment Neuron Features

Unsupervised sentiment neuron model [Radford et al., 2017] provides an
excellent learning representation of sentiment, despite being trained only on
the text of Amazon reviews. A linear model using this representation results
in good accuracy. This model represents a 4096 feature vector for any given
input tweet or text.

##  6 Experimental Setup & Results

To train our proposed approach, we consider a total of five learning models,
one for each expert: Gradient Boosting, XGBoost, Light Gradient Boosting,
Random Forest, and Neural Network(NN) for subtasks EI-reg and V-reg. While for
the subtasks EI-oc and V-oc, we consider all the models except NN model. For
subtask E-c, we consider all the models except Light Gradient Boosting model.

Model | Parameters
---|---
| n_estimators: 3000,
Gradient Boosting | Learning rate: 0.05
| Max_depth: 4
| n_estimators: 100
XGBoosting | learning_rate: 0.1
| max_depth: 3
| Optimizer: adam
Neural Network | Activation : relu
| n_estimators: 250
Random Forest | max_depth: 4
| n_estimators: 720
Light Gradient Boosting | learning_rate: 0.05
| num_leaves: 5
Table 2: Model-Parameters | EI-reg (Pearson (all instances)) | EI-reg (Pearson (gold in 0.5-1))
---|---|---
Team | macro-avg | anger | fear | joy | sadness | macro-avg | anger | fear | joy | sadness
SeerNet | 0.799(1) | 0.827 | 0.779 | 0.792 | 0.798 | 0.638(1) | 0.708 | 0.608 | 0.708 | 0.608
NTUA-SLP | 0.776(2) | 0.782 | 0.758 | 0.771 | 0.792 | 0.610(2) | 0.636 | 0.595 | 0.636 | 0.595
PlusEmo2Vec | 0.766(3) | 0.811 | 0.728 | 0.773 | 0.753 | 0.579(5) | 0.663 | 0.497 | 0.663 | 0.497
psyML | 0.765(4) | 0.788 | 0.748 | 0.761 | 0.761 | 0.593(4) | 0.657 | 0.541 | 0.657 | 0.541
Experts Model |  0.753(5) |  0.789 |  0.742 |  0.748 |  0.733 |  0.598(3) |  0.656 |  0.582 |  0.546 |  0.608
Median Team | 0.653(23) | 0.654 | 0.672 | 0.648 | 0.635 | 0.490(23) | 0.526 | 0.497 | 0.420 | 0.517
Baseline | 0.520(37) | 0.526 | 0.525 | 0.575 | 0.453 | 0.396(37) | 0.455 | 0.302 | 0.476 | 0.350
Note : The numbers inside parenthesis in both macro-avg columns represent the rank |  |
Table 3: Comparison of Regression results of various models with our Experts Model | EI-oc (Pearson (all classes)) | EI-oc (Pearson (some emotion))
---|---|---
Team | macro-avg | anger | fear | joy | sadness | macro-avg | anger | fear | joy | sadness
SeerNet | 0.695(1) | 0.706 | 0.637 | 0.720 | 0.717 | 0.547(1) | 0.559 | 0.458 | 0.610 | 0.560
PlusEmo2Vec | 0.659(2) | 0.704 | 0.528 | 0.720 | 0.683 | 0.501(4) | 0.548 | 0.320 | 0.604 | 0.533
psyML | 0.653(3) | 0.670 | 0.588 | 0.686 | 0.667 | 0.505(3) | 0.517 | 0.468 | 0.570 | 0.463
Amobee | 0.646(4) | 0.667 | 0.536 | 0.705 | 0.673 | 0.480(5) | 0.458 | 0.367 | 0.603 | 0.493
Experts Model |  0.636(5) |  0.658 |  0.576 |  0.666 |  0.644 |  0.520(2) |  0.493 |  0.502 |  0.579 |  0.509
Median Team | 0.530(17) | 0.530 | 0.470 | 0.552 | 0.567 | 0.415(17) | 0.408 | 0.310 | 0.494 | 0.448
Baseline | 0.394(26) | 0.382 | 0.355 | 0.469 | 0.370 | 0.296(26) | 0.315 | 0.183 | 0.396 | 0.289
Note : The numbers inside parenthesis in both macro-avg columns represent the rank |  |
Table 4: Comparison of Classification results of various models with our
Experts Model

###  6.1 Training Strategy

At the input layer, we used a concatenation vector of all features: Deep-
Emoji, Skip-Thought, Lexicons, Stylometric, BoW, Tf-IDF, Glove, Word2Vec,
Edinburgh, and HashTagIntensity which is same for each expert. We combined
both training and dev data and used them for training our model. The training
model is validated by stratified K-fold approach in which the model is
repeatedly trained on K-1 folds and the remaining one fold is used for
validation.

In order to tune the hyper-parameters of our experts model, we adopt a grid
search cross-validation for each learning model. Using grid search cross-
validation, we set the various types of parameters based on the learning
model. Table 2 shows the parameter settings for all experts.

###  6.2 Results

| V-reg (Pearson)
---|---
Team | (all instances) | (gold in 0.5-1)
SeerNet | 0.873 | 0.697
TCS Research | 0.861 | 0.680
PlusEmo2Vec | 0.860 | 0.691
NTUA-SLP | 0.851 | 0.688
Amobee | 0.843 | 0.644
Experts Model | 0.830 | 0.670
Median Team | 0.784 | 0.509
Baseline | 0.585 | 0.449
Table 5: Comparison of Valence-reg results of various models with our Experts
Model

| V-oc (Pearson)
---|---
Team | (all instances) | (gold in 0.5-1)
SeerNet | 0.836 | 0.884
PlusEmo2Vec | 0.833 | 0.878
Amobee | 0.813 | 0.865
psyML | 0.802 | 0.869
EiTAKA | 0.796 | 0.838
Experts Model | 0.738 | 0.773
Median Team | 0.682 | 0.754
Baseline | 0.509 | 0.560
Table 6: Comparison of Valence-oc results of various models with our Experts
Model

| E-c
---|---
Team | (acc.) | (micro F1) | (macro F1)
NTUA-SLP | 0.588(1) | 0.701 | 0.528
TCS Research | 0.582(2) | 0.693 | 0.530
PlusEmo2Vec | 0.576(4) | 0.692 | 0.497
psyML | 0.574(5) | 0.697 | 0.574
Experts Model | 0.578(3) | 0.691 | 0.581
Median Team | 0.471(17) | 0.599 | 0.464
Baseline | 0.442(21) | 0.570 | 0.443
Note : The numbers inside parenthesis in
accuracy column represent the rank
Table 7: Comparison of E-c results of various models with our Experts Model

To evaluate our computational model, we compare our results with SemEval-2018
Task-1 (Affect in Tweets) baseline results, top five performers and Median
Team (as per SemEval-2018 results). The results in the EI-reg, EI-oc, V-reg,
V-oc, E-c are shown in Tables 3, 4, 6, 6, 7 respectively. The tables
illustrate (a) the results obtained by our proposed approach, (b) top five
performers in SemEval-2018, (c) the results obtained by a baseline SVM system
using unigrams as features and (d) Median Team among all submissions. From the
Tables 3 and 4, we observe that our model (considering only macro-average for
Pearson Correlation) for EI-reg and EI-oc stands within 5 places among 48
submissions. A quick walk-through of Table 3 for individual emotions shows
that anger and fear ranks $3^{rd}$ and $4^{th}$ respectively for EI-reg
Pearson(all instances) and for EI-reg Pearson(gold in 0.5-1), fear stands at
$3^{rd}$ position and sadness equals score of top performer. Similarly, Table
4 for classification results shows that anger and fear ranks $4^{th}$ and
$3^{rd}$ respectively for EI-oc Pearson(all classes) and for EI-oc
Pearson(some emotion), anger, fear, joy and sadness stands at positions
$4^{th}$, $1^{st}$, $4^{th}$ and $3^{rd}$ respectively. It is to be noted that
in both Tables 3 and 4, the numbers inside parenthesis under column “macro-
avg” represent the rank according to macro-avg Pearson scores. These values
shows that our model stands at $3^{rd}$ and $2^{nd}$ positions in EI-reg
Pearson(gold in 0.5-1) and EI-oc Pearson(some emotion) respectively. Tables 6
and 6 illustrate that the results from our model are among the top 10
submissions of subtasks V-reg and V-oc. Table 7 shows the results of multi-
label emotion classification (11 classes). Our model is among the top 3
submissions for Jaccard similarity (accuracy) metric, in top 5 for micro F1
metric and topped the submissions for macro F1 metric.

###  6.3 Metrics

(a) Emotion: Anger

(b) Emotion: Fear

(c) Emotion: Joy

(d) Emotion: Sadness

Figure 2: Feature Importance: Comparison of Pearson Scores for each feature
vector & concatenated vector

We use the competition metric, Pearson Correlation Coefficient with the Gold
ratings/labels from SemEval-2018 task-1 AIT for EI-reg, EI-oc, V-reg and V-oc.
Further, macro-average was calculated by averaging the correlation scores of
four emotions: anger, fear, joy, and sadness for the tasks EI-reg and EI-oc.
Along with Pearson Correlation Coefficient, we use some additional metrics for
each subtask. The additional metric used for EI-reg and V-reg tasks was to
calculate the Pearson correlation only for a subset of test samples where the
intensity score was greater than or equal to 0.5. For the classification
subtasks EI-oc and V-oc, we use the additional metric Pearson correlation
calculated only for some emotion like low emotion, moderate emotion, or high
emotion. However, for the multi label emotion classification E-c, we used the
official evaluation metrics Jaccard Similarity (accuracy), micro average F1
score and macro average F1 score of all the classes.

Figure 2 shows the influence of each feature type on scores for predicting the
intensity or emotion. We can observe from Figure 2 that for “Deep-Emoji” and
“Deep-Emoji-Softmax” features, Pearson scores are dominating other feature
types. Feature types - Skip-Thought, Lexicons, Glove, and Edinburgh features
are contributing approximately similar in each of the 4 emotions. However,
Stylometric features and features from Unsupervised sentiment neurons are
performing worse.

##  7 Conclusion

In this paper, we have proposed a novel approach inspired from standard
Mixture-of-Experts model to predict the intensity of an emotion(Regression) or
level of an emotion (Classification) or multi-label emotion classification.
Experiment results show that our proposed approach can effectively deal the
emotion detection problem and stands at top-5 when compare with SemEval-2018
Task-1 AIT results and baseline results. As most of the Pearson scores are in
the range of 0.50 to 0.75, there is a lot of scope for improvement in
predicting emotions or quantifying the emotion intensity through various other
approaches, which are yet to be unfolded. The source code is publicly
available at https://goo.gl/NktJhF so that researchers and developers can work
on this exciting problem collectively.

## References

  * [Anchiêta et al., 2015]  Rafael T Anchiêta, Francisco Assis Ricarte Neto, Rogério Figueiredo de Sousa, and Raimundo Santos Moura.  2015\.  Using stylometric features for sentiment classification.  In International Conference on Intelligent Text Processing and Computational Linguistics, pages 189–200. Springer.
  * [Asur and Huberman, 2010]  Sitaram Asur and Bernardo A Huberman.  2010\.  Predicting the future with social media.  In Proceedings of the 2010 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology-Volume 01, pages 492–499. IEEE Computer Society.
  * [Baccianella et al., 2010]  Stefano Baccianella, Andrea Esuli, and Fabrizio Sebastiani.  2010\.  Sentiwordnet 3.0: an enhanced lexical resource for sentiment analysis and opinion mining.  In Lrec, volume 10, pages 2200–2204.
  * [Bauman et al., 2017]  Konstantin Bauman, Bing Liu, and Alexander Tuzhilin.  2017\.  Aspect based recommendations: Recommending items with the most valuable aspects based on user reviews.  In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 717–725. ACM.
  * [Bojanowski et al., 2017]  Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov.  2017\.  Enriching word vectors with subword information.  Transactions of the Association for Computational Linguistics, 5:135–146.
  * [dos Santos and Gatti, 2014]  Cicero dos Santos and Maira Gatti.  2014\.  Deep convolutional neural networks for sentiment analysis of short texts.  In Proceedings of COLING 2014, the 25th International Conference on Computational Linguistics: Technical Papers, pages 69–78.
  * [Duppada and Hiray, 2017]  Venkatesh Duppada and Sushant Hiray.  2017\.  Seernet at emoint-2017: Tweet emotion intensity estimator.  arXiv preprint arXiv:1708.06185.
  * [Duppada et al., 2018]  Venkatesh Duppada, Royal Jain, and Sushant Hiray.  2018\.  Seernet at semeval-2018 task 1: Domain adaptation for affect in tweets.  In Proceedings of The 12th International Workshop on Semantic Evaluation, SemEval@NAACL-HLT, New Orleans, Louisiana, June 5-6, 2018, pages 18–23.
  * [Felbo et al., 2017]  Bjarke Felbo, Alan Mislove, Anders Søgaard, Iyad Rahwan, and Sune Lehmann.  2017\.  Using millions of emoji occurrences to learn any-domain representations for detecting sentiment, emotion and sarcasm.  In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 1615–1625.
  * [Go et al., 2009]  Alec Go, Richa Bhayani, and Lei Huang.  2009\.  Twitter sentiment classification using distant supervision.  CS224N Project Report, Stanford, 1(12).
  * [Goel et al., 2017]  Pranav Goel, Devang Kulshreshtha, Prayas Jain, and Kaushal Kumar Shukla.  2017\.  Prayas at emoint 2017: An ensemble of deep neural architectures for emotion intensity prediction in tweets.  In Proceedings of the 8th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 58–65.
  * [Jacobs et al., 1991]  Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geoffrey E Hinton.  1991\.  Adaptive mixtures of local experts.  Neural Computation, 3(1):79–87.
  * [Jin et al., 2007]  Xin Jin, Ying Li, Teresa Mah, and Jie Tong.  2007\.  Sensitive webpage classification for content advertising.  In Proceedings of the 1st International Workshop on Data Mining and Audience Intelligence for Advertising, pages 28–33. ACM.
  * [Jordan and Jacobs, 1994]  Michael I Jordan and Robert A Jacobs.  1994\.  Hierarchical mixtures of experts and the em algorithm.  Neural Computation, 6(2):181–214.
  * [Jordan and Xu, 1995]  Michael I Jordan and Lei Xu.  1995\.  Convergence results for the em approach to mixtures of experts architectures.  Neural Networks, 8(9):1409–1431.
  * [Kiros et al., 2015]  Ryan Kiros, Yukun Zhu, Ruslan R Salakhutdinov, Richard Zemel, Raquel Urtasun, Antonio Torralba, and Sanja Fidler.  2015\.  Skip-thought vectors.  In Advances in Neural Information Processing Systems, pages 3294–3302.
  * [Köper et al., 2017]  Maximilian Köper, Evgeny Kim, and Roman Klinger.  2017\.  Ims at emoint-2017: emotion intensity prediction with affective norms, automatically extended resources and deep learning.  In Proceedings of the 8th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 50–57.
  * [Liu and Zhang, 2012]  Bing Liu and Lei Zhang.  2012\.  A survey of opinion mining and sentiment analysis.  In Mining Text Data, pages 415–463. Springer.
  * [Marsella et al., 2010]  Stacy Marsella, Jonathan Gratch, Paolo Petta, et al.  2010\.  Computational models of emotion.  A Blueprint for Affective Computing-A Sourcebook and Manual, 11(1):21–46.
  * [Mikolov et al., 2013]  Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean.  2013\.  Distributed representations of words and phrases and their compositionality.  In Advances in Neural Information Processing Systems, pages 3111–3119.
  * [Mohammad and Bravo-Marquez, 2017a]  Saif Mohammad and Felipe Bravo-Marquez.  2017a.  Emotion intensities in tweets.  In Proceedings of the 6th Joint Conference on Lexical and Computational Semantics (* SEM 2017), pages 65–77.
  * [Mohammad and Bravo-Marquez, 2017b]  Saif Mohammad and Felipe Bravo-Marquez.  2017b.  Wassa-2017 shared task on emotion intensity.  In Proceedings of the 8th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 34–49. Association for Computational Linguistics.
  * [Mohammad and Kiritchenko, 2015]  Saif M Mohammad and Svetlana Kiritchenko.  2015\.  Using hashtags to capture fine emotion categories from tweets.  Computational Intelligence, 31(2):301–326.
  * [Mohammad et al., 2018]  Saif M. Mohammad, Felipe Bravo-Marquez, Mohammad Salameh, and Svetlana Kiritchenko.  2018\.  Semeval-2018 Task 1: Affect in tweets.  In Proceedings of International Workshop on Semantic Evaluation (SemEval-2018), New Orleans, LA, USA.
  * [Mohammad, 2016]  Saif M Mohammad.  2016\.  Sentiment analysis: Detecting valence, emotions, and other affectual states from text.  In Emotion Measurement, pages 201–237. Elsevier.
  * [Nielsen, 2011]  Finn Årup Nielsen.  2011\.  A new anew: Evaluation of a word list for sentiment analysis in microblogs.  arXiv preprint arXiv:1103.2903.
  * [Nowlan and Hinton, 1991]  Steven J Nowlan and Geoffrey E Hinton.  1991\.  Evaluation of adaptive mixtures of competing experts.  In Advances in Neural Information Processing Systems, pages 774–780.
  * [Paltoglou et al., 2010]  Georgios Paltoglou, Mike Thelwall, and Kevan Buckley.  2010\.  Online textual communications annotated with grades of emotion strength.  In Proceedings of the 3rd International Workshop of Emotion: Corpora for research on Emotion and Affect, pages 25–31.
  * [Pang et al., 2008]  Bo Pang, Lillian Lee, et al.  2008\.  Opinion mining and sentiment analysis.  Foundations and Trends® in Information Retrieval, 2(1–2):1–135.
  * [Pennington et al., 2014]  Jeffrey Pennington, Richard Socher, and Christopher D. Manning.  2014\.  Glove: Global vectors for word representation.  In Empirical Methods in Natural Language Processing (EMNLP), pages 1532–1543.
  * [Petrović et al., 2010]  Saša Petrović, Miles Osborne, and Victor Lavrenko.  2010\.  The edinburgh twitter corpus.  In Proceedings of the NAACL HLT 2010 Workshop on Computational Linguistics in a World of Social Media, pages 25–26.
  * [Qiu et al., 2010]  Guang Qiu, Xiaofei He, Feng Zhang, Yuan Shi, Jiajun Bu, and Chun Chen.  2010\.  Dasa: dissatisfaction-oriented advertising based on sentiment analysis.  Expert Systems with Applications, 37(9):6182–6191.
  * [Radford et al., 2017]  Alec Radford, Rafal Jozefowicz, and Ilya Sutskever.  2017\.  Learning to generate reviews and discovering sentiment.  arXiv preprint arXiv:1704.01444.
  * [Somasundaran et al., 2007]  Swapna Somasundaran, Theresa Wilson, and Janyce Wiebe.  2007\.  Qa with attitude: Exploiting opinion type analysis for improving question answering in on-line discussions and the news.  In ICWSM.
  * [Staiano and Guerini, 2014]  Jacopo Staiano and Marco Guerini.  2014\.  Depeche mood: a lexicon for emotion analysis from crowd annotated news.  In Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), volume 2, pages 427–433.
  * [Stelmack and Stalikas, 1991]  Robert M Stelmack and Anastasios Stalikas.  1991\.  Galen and the humour theory of temperament.  Personality and Individual Differences, 12(3):255–263.
  * [Stoyanov et al., 2005]  Veselin Stoyanov, Claire Cardie, and Janyce Wiebe.  2005\.  Multi-perspective question answering using the opqa corpus.  In Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Language Processing, pages 923–930. Association for Computational Linguistics.
  * [Warriner et al., 2013]  Amy Beth Warriner, Victor Kuperman, and Marc Brysbaert.  2013\.  Norms of valence, arousal, and dominance for 13,915 english lemmas.  Behavior Research Methods, 45(4):1191–1207.
  * [Wilson et al., 2004]  Theresa Wilson, Janyce Wiebe, and Rebecca Hwa.  2004\.  Just how mad are you? finding strong and weak opinion clauses.  In Proceedings of the 19th National Conference on Artifical intelligence, pages 761–767.
  * [Wilson et al., 2005]  Theresa Wilson, Janyce Wiebe, and Paul Hoffmann.  2005\.  Recognizing contextual polarity in phrase-level sentiment analysis.  In Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Language Processing, pages 347–354. Association for Computational Linguistics.
  * [Yuksel et al., 2012]  Seniha Esen Yuksel, Joseph N Wilson, and Paul D Gader.  2012\.  Twenty years of mixture of experts.  IEEE Transactions on Neural Networks and Learning Systems, 23(8):1177–1193.