#  On FK’ Conjecture

Zhengtang Tan, Shouchuan Zhang, Weicai Wu
Department of Mathematics, Hunan University
Changsha 410082, P.R. China, Emails<EMAIL_ADDRESS>
###### Abstract

We give the relationship between FK’ Conjecture and Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ of transposition
over symmetry group by means of quiver Hopf algebras. That is, if
$\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)=\infty$, then
so is $\dim\mathcal{E}_{n}$.

2000 Mathematics Subject Classification: 16W30, 16G10

keywords: Conjecture, Hopf algebra, Weyl group.

##  0 Introduction

S. Fomin and A.N. Kirillov [FK97, Conjecture 2.2] point out a conjecture “
${\mathcal{E}}_{n}$ is finite dimensional ” to study the cohomology ring of
the flag manifold. In [FK] it is shown that ${\mathcal{E}}_{3}=12$ and
${\mathcal{E}}_{4}=24^{2}$ (see [AS02, Section 3.4]). Many papers ( for
example, [MS, GHV, Ba06]) refer to this conjecture.

In this paper we give the relationship between FK’ Conjecture and Nichols
algebra $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ of
transposition over symmetry group by means of quiver Hopf algebras. That is,
if $\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)=\infty$,
then so is $\dim\mathcal{E}_{n}$.

## Preliminaries And Conventions

Let ${k}$ be the complex field, A quiver $Q=(Q_{0},Q_{1},s,t)$ is an oriented
graph, where $Q_{0}$ and $Q_{1}$ are the sets of vertices and arrows,
respectively; $s$ and $t$ are two maps from $Q_{1}$ to $Q_{0}$. For any arrow
$a\in Q_{1}$, $s(a)$ and $t(a)$ are called its start vertex and end vertex,
respectively, and $a$ is called an arrow from $s(a)$ to $t(a)$. For any $n\geq
0$, an $n$-path or a path of length $n$ in the quiver $Q$ is an ordered
sequence of arrows $p=a_{n}a_{n-1}\cdots a_{1}$ with $t(a_{i})=s(a_{i+1})$ for
all $1\leq i\leq n-1$. Note that a 0-path is exactly a vertex and a 1-path is
exactly an arrow. In this case, we define $s(p)=s(a_{1})$, the start vertex of
$p$, and $t(p)=t(a_{n})$, the end vertex of $p$. For a 0-path $x$, we have
$s(x)=t(x)=x$. Let $Q_{n}$ be the set of $n$-paths. Let ${}^{y}Q_{n}^{x}$
denote the set of all $n$-paths from $x$ to $y$, $x,y\in Q_{0}$. That is,
${}^{y}Q_{n}^{x}=\\{p\in Q_{n}\mid s(p)=x,t(p)=y\\}$.

A quiver $Q$ is finite if $Q_{0}$ and $Q_{1}$ are finite sets. A quiver $Q$ is
locally finite if ${}^{y}Q_{1}^{x}$ is a finite set for any $x,y\in Q_{0}$.

Let ${\mathcal{K}}(G)$ denote the set of conjugacy classes in $G$. A formal
sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugacy classes of $G$ with
cardinal number coefficients is called a ramification (or ramification data )
of $G$, i.e. for any $C\in{\mathcal{K}}(G)$, $r_{C}$ is a cardinal number. In
particular, a formal sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugacy
classes of $G$ with non-negative integer coefficients is a ramification of
$G$.

For any ramification $r$ and $C\in{\mathcal{K}}(G)$, since $r_{C}$ is a
cardinal number, we can choose a set $I_{C}(r)$ such that its cardinal number
is $r_{C}$ without loss of generality. Let
${\mathcal{K}}_{r}(G):=\\{C\in{\mathcal{K}}(G)\mid
r_{C}\not=0\\}=\\{C\in{\mathcal{K}}(G)\mid I_{C}(r)\not=\emptyset\\}$. If
there exists a ramification $r$ of $G$ such that the cardinal number of
${}^{y}Q_{1}^{x}$ is equal to $r_{C}$ for any $x,y\in G$ with $x^{-1}y\in
C\in{\mathcal{K}}(G)$, then $Q$ is called a Hopf quiver with respect to the
ramification data $r$. In this case, there is a bijection from $I_{C}(r)$ to
${}^{y}Q_{1}^{x}$, and hence we write ${\ }^{y}Q_{1}^{x}=\\{a_{y,x}^{(i)}\mid
i\in I_{C}(r)\\}$ for any $x,y\in G$ with $x^{-1}y\in C\in{\mathcal{K}}(G)$.

$(G,r,\overrightarrow{\rho},u)$ is called a ramification system with
irreducible representations (or RSR in short ), if $r$ is a ramification of
$G$; $u$ is a map from ${\mathcal{K}}(G)$ to $G$ with $u(C)\in C$ for any
$C\in{\mathcal{K}}(G)$; $I_{C}(r,u)$ and $J_{C}(i)$ are sets with
$\mid\\!J_{C}(i)\\!\mid$ = ${\rm deg}(\rho_{C}^{(i)})$ and
$I_{C}(r)=\\{(i,j)\mid i\in I_{C}(r,u),j\in J_{C}(i)\\}$ for any
$C\in{\mathcal{K}}_{r}(G)$, $i\in I_{C}(r,u)$;
$\overrightarrow{\rho}=\\{\rho_{C}^{(i)}\\}_{i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)}\
\in\prod_{C\in{\mathcal{K}}_{r}(G)}(\widehat{{G^{u(C)}}})^{\mid
I_{C}(r,u)\mid}$ with $\rho_{C}^{(i)}\in\widehat{{G^{u(C)}}}$ for any $i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)$. In this paper we always assume that
$I_{C}(r,u)$ is a finite set for any $C\in{\mathcal{K}}_{r}(G).$ Furthermore,
if $\rho_{C}^{(i)}$ is a one dimensional representation for any
$C\in{\mathcal{K}}_{r}(G)$, then $(G,r,\overrightarrow{\rho},u)$ is called a
ramification system with characters  (or RSC $(G,r,\overrightarrow{\rho},u)$
in short ) (see [ZZC04, Definition 1.8]). In this case, $a_{y,x}^{(i,j)}$ is
written as $a_{y,x}^{(i)}$ in short since $J_{C}(i)$ has only one element.

For ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, let $\chi_{C}^{(i)}$ denote the
character of $\rho_{C}^{(i)}$ for any $i\in I_{C}(r,u)$,
$C\in{\mathcal{K}}_{r}(C)$. If ramification $r=r_{C}C$ and
$I_{C}(r,u)=\\{i\\}$ then we say that ${\rm RSR}(G,r,\overrightarrow{\rho},u)$
is bi-one, written as ${\rm RSR}(G,{\mathcal{O}}_{s},\rho)$ with $s=u(C)$ and
$\rho=\rho_{C}^{(i)}$ in short, since $r$ only has one conjugacy class $C$ and
$\mid\\!I_{C}(r,u)\\!\mid=1$. Quiver Hopf algebras, Nichols algebras and
Yetter-Drinfeld modules, corresponding to a bi-one ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$, are said to be bi-one.

If $(G,r,\overrightarrow{\rho},u)$ is an ${\rm RSR}$, then it is clear that
${\rm RSR}(G,{\mathcal{O}}_{u(C)},\rho_{C}^{(i)})$ is bi-one for any
$C\in{\mathcal{K}}$ and $i\in I_{C}(r,u)$, which is called a bi-one  sub-${\rm
RSR}$ of ${\rm RSR}(G,r,\overrightarrow{\rho},u)$,

For $s\in G$ and $(\rho,V)\in\widehat{G^{s}}$, here is a precise description
of the YD module $M({\mathcal{O}}_{s},\rho)$, introduced in [Gr00, AZ07]. Let
$t_{1}=s$, …, $t_{m}$ be a numeration of ${\mathcal{O}}_{s}$, which is a
conjugacy class containing $s$, and let $g_{i}\in G$ such that $g_{i}\rhd
s:=g_{i}sg_{i}^{-1}=t_{i}$ for all $1\leq i\leq m$. Then
$M({\mathcal{O}}_{s},\rho)=\oplus_{1\leq i\leq m}g_{i}\otimes V$. Let
$g_{i}v:=g_{i}\otimes v\in M({\mathcal{O}}_{s},\rho)$, $1\leq i\leq m$, $v\in
V$. If $v\in V$ and $1\leq i\leq m$, then the action of $h\in G$ and the
coaction are given by

$\displaystyle\delta(g_{i}v)=t_{i}\otimes g_{i}v,\qquad
h\cdot(g_{i}v)=g_{j}(\gamma\cdot v),$ (0.1)

where $hg_{i}=g_{j}\gamma$, for some $1\leq j\leq m$ and $\gamma\in G^{s}$.
The explicit formula for the braiding is then given by

$c(g_{i}v\otimes g_{j}w)=t_{i}\cdot(g_{j}w)\otimes
g_{i}v=g_{j^{\prime}}(\gamma\cdot w)\otimes g_{i}v$ (0.2)

for any $1\leq i,j\leq m$, $v,w\in V$, where $t_{i}g_{j}=g_{j^{\prime}}\gamma$
for unique $j^{\prime}$, $1\leq j^{\prime}\leq m$ and $\gamma\in G^{s}$. Let
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ denote
$\mathfrak{B}(M({\mathcal{O}}_{s},\rho))$. $M({\mathcal{O}}_{s},\rho)$ is a
simple YD module (see [AZ07, Section 1.2 ]).

##  1 Relation between bi-one arrow Nichols algebras and
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ (see [WZT13])

In this section it is shown that bi-one arrow Nichols algebras and
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ introduced in [Gr00, AZ07, AFZ] are the
same up to isomorphisms.

For any ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, we can construct an arrow
Nichols algebra $\mathfrak{B}(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},$ $u))$
( see [ZCZ, Pro. 2.4]), written as $\mathfrak{B}(G,r,\overrightarrow{\rho},$
$u)$ in short. Let us recall the precise description of arrow YD module. For
an ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and a $kG$-Hopf bimodule
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ with the module operations
$\alpha^{-}$ and $\alpha^{+}$, define a new left $kG$-action on $kQ_{1}$ by

$g\rhd x:=g\cdot x\cdot g^{-1},\ g\in G,x\in kQ_{1},$

where $g\cdot x=\alpha^{-}(g\otimes x)$ and $x\cdot g=\alpha^{+}(x\otimes g)$
for any $g\in G$ and $x\in kQ_{1}$. With this left $kG$-action and the
original left (arrow) $kG$-coaction $\delta^{-}$, $kQ_{1}$ is a Yetter-
Drinfeld $kG$-module. Let $Q_{1}^{1}:=\\{a\in Q_{1}\mid s(a)=1\\}$, the set of
all arrows with starting vertex $1$. It is clear that $kQ_{1}^{1}$ is a
Yetter-Drinfeld $kG$-submodule of $kQ_{1}$, denoted by
$(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$, called the arrow YD module.

######  Lemma 1.1.

For any $s\in G$ and $\rho\in\widehat{G^{s}}$, there exists a bi-one arrow
Nichols algebra $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ such that

$\mathfrak{B}({\mathcal{O}}_{s},\rho)\cong\mathfrak{B}(G,r,\overrightarrow{\rho},u)$

as graded braided Hopf algebras in ${}^{kG}_{kG}\\!{\mathcal{Y}D}$.

Proof. Assume that $V$ is the representation space of $\rho$ with
$\rho(g)(v)=g\cdot v$ for any $g\in G,v\in V$. Let $C={\mathcal{O}_{s}}$,
$r=r_{C}C$, $r_{C}={\rm deg}\rho$, $u(C)=s$, $I_{C}(r,u)=\\{1\\}$ and
$(v)\rho_{C}^{(1)}(h)=\rho(h^{-1})(v)$ for any $h\in G$, $v\in V$. We get a
bi-one arrow Nichols algebra $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$.

We now only need to show that
$M({\mathcal{O}}_{s},\rho)\cong(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$
in ${}^{kG}_{kG}\\!{\mathcal{Y}D}$. We recall the notation in [ZCZ,
Proposition 1.2]. Assume $J_{C}(1)=\\{1,2,\cdots,n\\}$ and $X_{C}^{(1)}=V$
with basis $\\{x_{C}^{(1,j)}\mid j=1,2,\cdots,n\\}$ without loss of
generality. Let $v_{j}$ denote $x_{C}^{(1,j)}$ for convenience. In fact, the
left and right coset decompositions of $G^{s}$ in $G$ are

$\displaystyle G=\bigcup_{i=1}^{m}g_{i}G^{s}\ \ \hbox{and }\ \ G$
$\displaystyle=$ $\displaystyle\bigcup_{i=1}^{m}G^{s}g_{i}^{-1}\ \ ,$ (1.1)

respectively.

Let $\psi$ be a map from $M({\mathcal{O}}_{s},\rho)$ to $(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))$ by sending $g_{i}v_{j}$ to
$a_{t_{i},1}^{(1,j)}$ for any $1\leq i\leq m,1\leq j\leq n$. Since the
dimension is $mn$, $\psi$ is a bijective. See

$\displaystyle\delta^{-}(\psi(g_{i}v_{j}))$ $\displaystyle=$
$\displaystyle\delta^{-}(a_{t_{i},1}^{(1,j)})$ $\displaystyle=$ $\displaystyle
t_{i}\otimes a_{t_{i},1}^{(1,j)}=(id\otimes\psi)\delta^{-}(g_{i}v_{j}).$

Thus $\psi$ is a $kG$-comodule homomorphism. For any $h\in G$, assume
$hg_{i}=g_{i^{\prime}}\gamma$ with $\gamma\in G^{s}$. Thus
$g_{i}^{-1}h^{-1}=\gamma^{-1}g_{i^{\prime}}^{-1}$, i.e.
$\zeta_{i}(h^{-1})=\gamma^{-1}$, where $\zeta_{i}$ was defined in [ZZC04,
(0.3)]. Since $\gamma\cdot x^{(1,j)}\in V$, there exist
$k_{C,h^{-1}}^{(1,j,p)}\in k$, $1\leq p\leq n$, such that $\gamma\cdot
x^{(1,j)}=\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}x^{(1,p)}$. Therefore

$\displaystyle x^{(1,j)}\cdot\zeta_{i}(h^{-1})$ $\displaystyle=$
$\displaystyle\gamma\cdot x^{(1,j)}\ \ (\hbox{by definition of
}\rho_{C}^{(1)})$ (1.2) $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}x^{(1,p)}.$

See

$\displaystyle\psi(h\cdot g_{i}v_{j})$ $\displaystyle=$
$\displaystyle\psi(g_{i^{\prime}}(\gamma v_{j}))$ $\displaystyle=$
$\displaystyle\psi(g_{i^{\prime}}(\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}v_{p}))$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}a_{t_{i^{\prime}},1}^{(1,p)}$

and

$\displaystyle h\rhd(\psi(g_{i}v_{j}))$ $\displaystyle=$ $\displaystyle
h\rhd(a_{t_{i},1}^{(1,j)})$ $\displaystyle=$ $\displaystyle
a_{ht_{i},h}^{(1,j)}\cdot h^{-1}$ $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}a_{t_{i^{\prime}},1}^{(1,p)}\
\ (\hbox{by \cite[cite]{[\@@bibref{}{ZCZ08}{}{}, Pro.1.2]} and
}(\ref{e1.11})).$

Therefore $\psi$ is a $kG$-module homomorphism. $\Box$

Therefore we also say that $\mathfrak{B}({\mathcal{O}}_{s},\rho)$ is a bi-one
Nichols Hopf algebra.

######  Remark 1.2.

The representation $\rho$ in $\mathfrak{B}({\mathcal{O}}_{s},\rho)$ introduced
in [Gr00, AZ07] and $\rho_{C}^{(i)}$ in RSR are different. $\rho(g)$ acts on
its representation space from the left and $\rho_{C}^{(i)}(g)$ acts on its
representation space from the right.

Otherwise, when $\rho=\chi$ is a one dimensional representation, then
$(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$ is PM (see [ZZC04, Def. 1.1]).
Thus the formulae are available in [ZZC04, Lemma 1.9]. That is, $g\cdot
a_{t}=a_{gt_{i},g}$, $a_{t_{i}}\cdot g=\chi(\zeta_{i}(g))a_{t_{i}g,g}$.

##  2 Transposition and FK’ Conjecture

In this section we give the relationship between FK’ Conjecture and Nichols
algebra $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ of
transposition over symmetry group by means of quiver Hopf algebras.

We first consider Nichols algebra $\mathfrak{B}({\mathcal{O}}_{(12)},\rho)$ of
transposition $\sigma$ over symmetry groups, where $\rho=sgn\otimes sgn$ or
$\rho=\epsilon\otimes sgn.$

Let $\sigma=(12)\in S_{n}:=G$, $\mathcal{O}_{\sigma}=\\{(ij)|1\leq i,j\leq
n\\}$, $G^{\sigma}=\\{g\in G\mid g\sigma=\sigma g\\}$.
$G=\bigcup\limits_{1\leq i<j\leq n}G^{\sigma}g_{i,j}$.

Let $g_{kj}:=\left\\{\begin{array}[]{lll}id\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
k=1,j=2\\\ (2j)\ \ \ \ \ \ \ \ \ \ \ \ \ \ k=1,j>2\\\ (1j)\ \ \ \ \ \ \ \ \ \
\ \ \ \ k=2,j>2\\\ (1k)(2j)\ \ \ \ \ \ \ \ \ k>2,j>k\\\ \end{array}\right.$
and $t_{kj}:=(k\ j).$

######  Lemma 2.1.

(see [WZT13]) In The following equations in $\mathbb{S}_{m}$ hold.

$(12)id=id(12)$

$(12)(2j)=(2j1)=(1j)(12)$

$(12)(1j)=(1j2)=(2j)(12)$

$(12)(1k)(2j)=(1k2)(2j)=(1k)(2j)(kj)$

$(1j)id=(1j)id$

$(1j)(2j)=(j21)=(2j)(12)$

$(1j)(2j_{1})=(1j)(2j_{1})id\ \ j<j_{1}$

$(1j)(2j_{1})=(1j_{1})(2j)(jj_{1})(12)\ \ j>j_{1}$

$(1j)(1j)=id$

$(1j)(1j_{1})=(1j_{1}j)=(1j_{1})(jj_{1})$

$(1j)(1k)(2j)=(1kj)(2j)=(2k)(12)(kj)$

$(1j)(1k)(2j_{1})=(2j_{1})id\ \ \ j=k$

$(1j)(1k)(2j_{1})=(1k)(kj)(2j_{1})=(1k)(2j_{1})(kj)\ \ j\neq j_{1}$

$(2j)id=(2j)id$

$(2j)(2j)=id$

$(2j)(2j_{1})=(2j_{1}j)=(2j_{1})(jj_{1})$

$(2j)(1j)=(j12)=(1j)(12)$

$(2j)(1j_{1})=(1j)(2j_{1})(jj_{1})(12)\ \ j<j_{1}$

$(2j)(1j_{1})=(1j_{1})(2j)id\ \ j>j_{1}$

$(2j)(1k)(2j)=(1k)=(1k)id$

$(2j)(1k)(2j_{1})=(1k)(12)(2j_{1})=(1k)(1j_{1})(12)=(1j_{1})(12)(kj_{1})\ \
j=k$

$(2j)(1k)(2j_{1})=(2j)(2j_{1})(1k)=(2j_{1})(j_{1}j)(1k)=(2j_{1})(1k)(j_{1}j)\
\ j\neq j_{1},j\neq k$

$(kj)id=id(kj)$

$(kj)(2j)=(2k)(kj)$

$(kj)(2k)=(2j)(kj)$

$(kj)(2j_{1})=(2j_{1})(kj)\ \ k\neq j_{1},j\neq j_{1}$

$(kj)(1j)=(1k)(kj)$

$(kj)(1k)=(1j)(kj)$

$(kj)(1j_{1})=(1j_{1})(kj)\ \ k\neq j_{1},j\neq j_{1}$

$(kj)(1k)(2j)=(1k)(2j)(12)$

$(kj)(1k_{1})(2j_{1})=(1k)(2j_{1})(kj)\ \ k_{1}=j$

$(kj)(1k_{1})(2j_{1})=(1k_{1})(2k)(kj)\ \ k_{1}<k,j_{1}=j$

$(kj)(1k_{1})(2j_{1})=(1k)(2k_{1})(12)(kjk_{1})\ \ k_{1}>k,j_{1}=j$

$(kj)(1k_{1})(2j_{1})=(1j)(2j_{1})(kj)\ \ j_{1}>j,k_{1}=k$

$(kj)(1k_{1})(2j_{1})=(1j_{1})(2j)(12)(jkj_{1})\ \ j_{1}<j,k_{1}=k$

$(kj)(1k_{1})(2j_{1})=(1k_{1})(2j)(kj)\ \ k_{1}\neq j,j_{1}=k$

$(kj)(1k_{1})(2j_{1})=(1k_{1})(2j_{1})(kj)\ \ k_{1}\neq k,k_{1}\neq
j,j_{1}\neq j,j_{1}\neq k$.

Remark: By Lemma above, we can obtain $\zeta_{st}(t_{ij})\in G^{\sigma}$ such
that $g_{st}t_{ij}=\zeta_{st}(t_{ij})g_{s^{\prime}t^{\prime}}$ for any $1\leq
i,j,s,t\leq n.$ Let $a_{ij}$ denote the arrow $a_{t_{ij},1}$ from $1$ to
$t_{ij}$ in short. By Lemma 1.1, the algebra generated by $\\{a_{ij}\mid 1\leq
i<j\leq n\\}$ in its co-path Hopf algebra is isomorphic to Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho).$

######  Definition 2.2.

(See [FK97, Def. 2.1]) algebra ${\mathcal{E}}_{n}$ is generated by
$\\{x_{ij}\mid 1\leq i<j\leq n\\}$ with definition relations:

(i) $x_{ij}^{2}=0$ for $i<j.$

(ii) $x_{ij}x_{jk}=x_{jk}x_{ik}+x_{ik}x_{ij}$ and
$x_{jk}x_{ij}=x_{ik}x_{jk}+x_{ij}x_{ik},$ for $i<j<k.$

(iii) $x_{ij}x_{kl}=x_{kl}x_{ij}$ for any distinct $i,j,k$ and $l,$ $i<j,k<l.$

######  Theorem 2.3.

Assume $n>3$.

(i) $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ is an image of
$\mathcal{E}_{n}$.

(ii) If $\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes
sgn)=\infty$, then so is $\dim\mathcal{E}_{n}$.

Proof. Let $b_{ij}=-a_{ij}$ when $i=2$ and $j>2$; $b_{ij}=a_{ij}$ otherwise.
By [WZT13, Table 1, Lemma 5.2(i)(ii)(iii)], there exists $\alpha_{i,j,k}$,
$\beta_{i,j,k}\in\\{1,-1\\}$ such that

$\displaystyle
a_{ij}a_{jk}+\alpha_{ijk}a_{jk}a_{ki}+\beta_{ijk}a_{ki}a_{ij}=0,$ (2.1)

for any distinct $i,j$ and $k$. It follows from [WZT13, formula (5.2) ] that
case $\alpha_{ijk}$ $\beta_{ijk}$ $2<i<j<k$ $-1$ $-1$ $i=1,j=2<k$ $-1$ $1$
$i=1,2<j<k$ $-1$ $-1$ $i=2<j<k$ $-1$ $1$ $2<i<k<j$ $-1$ $1$ $i=1,k=2<j$ $-1$
$-1$ $i=1,2<k<j$ $-1$ $1$ $i=2<k<j$ $-1$ $-1$ .
$\hbox{Table }1$

Using the Table 1 we can obtain that $b_{ij}^{2}=0$ for $i<j;$
$b_{ij}b_{jk}=b_{jk}b_{ik}+b_{ik}b_{ij}$ and
$b_{jk}b_{ij}=b_{ik}b_{jk}+b_{ij}b_{ik},$ for $i<j<k.$ By [WZT13, Lemma
5.2(iv)], $b_{ij}b_{kl}=b_{kl}b_{ij}$ for any distinct $i,j,k$ and $l,$
$i<j,k<l.$ Consequently, Part (i) holds since $\\{b_{ij}\mid 1\leq i<j\leq
n\\}$ generates $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$.
$\Box$

Notation: N. Andruskiewitsch and H.-J. Schneider point out that
${\mathcal{E}}_{5}$ is finite dimensional by using a computer program (see
[AS02, Section 3.4]). A. Kirillov told me that ${\mathcal{E}}_{5}$ has
dimension $4^{4}5^{2}6^{4}$ and the real problem is to compute the Hilbert
series of the algebra ${\mathcal{E}}_{6}$. Consequently,
$\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)<\infty$ when
$n=5.$

Let $\chi^{\prime}:=sgn\otimes sgn$ and $\chi^{\prime\prime}:=\epsilon\otimes
sgn$. Let $\phi_{1}$ and $\phi_{2}$ be maps from $\mathbb{S}_{n}\times T$ to
${\bf k}\setminus 0$ such that
$\phi_{1}(g,t):=\left\\{\begin{array}[]{ll}1&\hbox{if }g(i)<g(j)\\\
-1&\hbox{if }g(j)<g(i)\\\ \end{array}\right.$ and $\phi_{2}(g,t):=(-1)^{l(g)}$
where $1\leq<j\leq n$, $T:=\\{(u,v)\mid 1\leq u,v\leq n,u\not=v\\}$ and $l(g)$
is the length of $g$, that is, $l(g)$ is the minimal number $q$ such that
$g=g_{1}g_{2}\cdots g_{q}$ with $g_{i}\in T$ , $1\leq i\leq q.$ Let
$\\{x_{ij}\mid(i,j)\in T\\}$ be a basis of $M(\mathbb{S}_{n},T,\phi)$ with
$\phi=\phi_{1}$ or $\phi=\phi_{2}.$ Define module and comodule operations as
follows: $g\rhd x_{ij}=\phi(g,(i,j))x_{g\rhd(i,j)}$,
$\delta^{-}(x_{ij})=(i,j)\otimes x_{ij},$ for any $g\in\mathbb{S}_{n},(i,j)\in
T.$ By [MS, Def. 5.1 and Example 5.3], $M(\mathbb{S}_{n},T,\phi)$ is a ${\rm
YD}$ module over $\mathbb{S}_{n}.$ Its Nichols algebra is written
$\mathfrak{B}(\mathbb{S}_{n},T,\phi)$. By [FK97, Def. 2.1] and [MS, Example
6.2 ], $\mathfrak{B}(\mathbb{S}_{n},T,\phi)$ is an image of $\mathcal{E}_{n}$.

######  Lemma 2.4.

If $n>4$, then $M(\mathbb{S}_{n},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime})$ as YD modules over
$\mathbb{S}_{n}.$ $M(\mathbb{S}_{m},T,\phi_{2})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime\prime})$ as YD modules over
$\mathbb{S}_{n}$.

Proof. If there exists isomorphism $\psi:M(\mathbb{S}_{n},T,\phi)\rightarrow
M({\mathcal{O}}_{{(1,2)}},\chi)$ as YD mosules over $\mathbb{S}_{n}$, where
$\phi=\phi_{1}$ or $\phi=\phi_{2}$, $\chi=\chi^{\prime}$ or
$\chi=\chi^{\prime\prime}$, then there exists $k_{ij}\in{\bf k}$ such that
$\psi(x_{ij})=k_{ij}a_{ij}$ for any $(i,j)\in T$, where $a_{ij}$ denotes arrow
$a_{t_{ij},1}$ in short, since $\psi$ is a comodule isomorphism. By
$\psi(g\rhd x_{ij})=g\rhd\psi(x_{ij})$, we have

$\displaystyle
k_{g\rhd(ij)}\phi(g,(i,j))a_{g\rhd(ij)}=k_{ij}\chi(\zeta_{ij}(g^{-1}))a_{g\rhd(ij)},$
(2.2)

for any $g\in\mathbb{S}_{n}$, $(i,j)\in T.$ For convenience, set
$k_{ij}=k_{ji}.$

Let $2<i<j$, $g=(1,2)$. It is clear $g_{ij}g=(1i)(2j)(12)=(ij)(1i)(2j)$. We
have $g=g^{-1}$ and $\zeta_{ij}(g)=(ij)$. Thus
$\chi^{\prime}(\zeta_{ij}(g))=-1$ and $\chi^{\prime\prime}(\zeta_{ij}(g))=1$;
$\phi_{1}(g,(ij))=1$ and $\phi_{2}(g,(ij))=-1$. Considering (2.2) we have
$M(\mathbb{S}_{n},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime})$ as YD modules over
$\mathbb{S}_{n}.$ $M(\mathbb{S}_{m},T,\phi_{2})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime\prime})$ as YD modules over
$\mathbb{S}_{n}$ since
$k_{g\rhd(ij)}\phi_{1}(g,(i,j))\not=k_{ij}\chi^{\prime}(\zeta_{ij}(g))$ and
$k_{g\rhd(ij)}\phi_{2}(g,(i,j))\not=k_{ij}\chi^{\prime\prime}(\zeta_{ij}(g))$.
$\Box$

######  Theorem 2.5.

If there exists a natural number $n_{0}>5$ such that
$M(\mathbb{S}_{n_{0}},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime\prime})$ as YD modules over
$\mathbb{S}_{n_{0}}$, then
$\dim\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})=\infty$ and
$\dim\mathcal{E}_{n}=\infty$ for any $n>n_{0}.$.

Proof. $M(\mathbb{S}_{6},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)(3,4)(5,6)}},\rho)$ as YD modules over
$\mathbb{S}_{6}$ since they are not isomorphic as comodules over
$\mathbb{S}_{6}$, when $\rho$ is one dimensional representation.

If $\dim\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})<\infty$, then
$\dim\mathfrak{B}(\mathbb{S}_{n_{0}},T,\phi_{1})<\infty$ since
$\mathbb{S}_{n_{0}}$ is a subgroup of $\mathbb{S}_{n}.$
$M(\mathbb{S}_{n},T,\phi_{1})$ is a reducible YD modules over $\mathbb{S}_{n}$
by Lemma 2.4 and [AFGV08, Th. 1.1]. However, every reducible YD modules over
$\mathbb{S}_{n}$ is infinite dimensional by [HS08, Cor. 8.4]. This is a
contradiction. Consequently,
$\dim\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})=\infty.$ By [FK97, Def. 2.1] and
[MS, Example 6.2 ], $\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})$ is an image of
$\mathcal{E}_{n}$. Therefore, $\dim\mathcal{E}_{n}=\infty$ for any $n>n_{0}.$.
$\Box$

## References

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  * [AFGV08] N. Andruskiewitsch, F. Fantino, M. Graña and L.Vendramin, Finite-dimensional pointed Hopf algebras with alternating groups are trivial, preprint arXiv:0812.4628, to appear Aparecer en Ann. Mat. Pura Appl..
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  * [AZ07] N. Andruskiewitsch and S. Zhang, On pointed Hopf algebras associated to some conjugacy classes in $\mathbb{S}_{n}$, Proc. Amer. Math. Soc. 135 (2007), 2723-2731.
  * [Ba06] Y. Bazlov, Nichols CWoronowicz algebra model for Schubert calculus on Coxeter groups, Journal of Algebra, 297 ( 2006) 2, 372 C399.
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  * [FK97] S. Fomin and A.N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Progress in Geometry, ed. J.-L. Brylinski and R. Brylinski, 1997.
  * [GHV] M. Graña, I. Heckenberger, L. Vendramin , Nichols algebras of group type with many quadratic relations, Advances in Mathematics, 227(2011)5, 1956-1989.
  * [Gr00] M. Graña, On Nichols algebras of low dimension, Contemp. Math. 267 (2000), 111–134.
  * [HS08] I. Heckenberger and H.-J. Schneider, Root systems and Weyl groupoids for Nichols algebras, preprint arXiv:0807.0691, to appear Proc. London Math. Soc..
  * [MS] A. Milinski and H-J. Schneider, _Pointed Indecomposable Hopf Algebras over Coxeter Groups_ , Contemp. Math. 267 (2000), 215–236.
  * [WZT13] Weicai Wu, Shouchuan Zhang, Zhengtang Tan, Pointed Hopf algebras with classical Weyl groups (II), Preprint arXiv: 1307.8227.
  * [ZZC04] Shouchuan Zhang, Y-Z Zhang and H. X. Chen, Classification of PM Quiver Hopf Algebras, J. Alg. and Its Appl. 6 (2007)(6), 919-950. Also see in math. QA/0410150.
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# Complexity of human response delay in intermittent control: The case of
virtual stick balancing

Takashi Suzuki<EMAIL_ADDRESS>School of Computer Science and
Engineering, University of Aizu, Japan  Ihor Lubashevsky
<EMAIL_ADDRESS>School of Computer Science and Engineering, University
of Aizu, Japan  Arkady Zgonnikov<EMAIL_ADDRESS>School of Computer
Science and Engineering, University of Aizu, Japan

###### Abstract

Response delay is an inherent and essential part of human actions. In the
context of human balance control, the response delay is traditionally modeled
using the formalism of delay-differential equations, which adopts the
approximation of fixed delay. However, experimental studies revealing
substantial variability, adaptive anticipation, and non-stationary dynamics of
response delay provide evidence against this approximation. In this paper, we
call for development of principally new mathematical formalism describing
human response delay. To support this, we present the experimental data from a
simple virtual stick balancing task. Our results demonstrate that human
response delay is a widely distributed random variable with complex
properties, which can exhibit oscillatory and adaptive dynamics characterized
by long-range correlations. Given this, we argue that the fixed-delay
approximation ignores essential properties of human response, and conclude
with possible directions for future developments of new mathematical notions
describing human control.

##  I Introduction

Human control often invokes discontinuous, intermittent mechanisms [1; 2; 3].
In car driving [4; 5], postural sway [6; 7], balancing a stick on a fingertip
[1; 8], etc., humans tend to hold off the control until the deviation of the
controlled system from the desired state is large enough, and only then apply
a corrective action. More continuous control strategies are less effective
compared to such intermittent corrections, in part due to significant human
response delays [9; 10], which can reach hundreds of milliseconds. Indeed,
throughout the control loop (stimulus perception, response execution), the
speed of signal propagation in the nervous system is finite [11; 12]. In
addition, responding to observed deviations has been shown to involve
stochastic decision-making mechanisms [5; 13], which require non-zero time to
generate a response. For these reasons, humans cannot immediately respond to
the input signal from the controlled system, and have to mitigate the effects
of the resulting lag via control intermittency or other compensatory
mechanisms [14; 15; 16; 17].

The crucial role of response delay in human intermittent control is now
universally acknowledged: virtually all relevant models capture the delay
effects in some way. By far the most popular approach adopts the formalism of
delay differential equations (DDE) (see [18; 19; 20] for general background,
and [21; 22] for discussion of response delay specifically in human balance
control). DDE have a number of advantages compared to more sophisticated
approaches (e.g., equations with distributed delay); most importantly, DDE can
be efficiently solved numerically. At the same time, there is evidence that
traditional DDE, albeit being useful in understanding human control, do not
provide an accurate representation of human response delay.

The main drawback of the DDE-based models is that they effectively assume that
human response delay is fixed, and cannot account for delay variability, which
is observed experimentally in balancing literature [23; 17]. Furthermore, even
such experimental studies of response delay in human balance control tend to
focus on average delay values, putting little emphasis on delay variability.
In a recent study of stick balancing on a fingertip, Milton et al. [10]
estimated the delay to be 230 ms based on a task with a sensory blank-out, but
the method used there required averaging across trials and therefore did not
allow to estimate the delay variability. Even when full distributions of
observed delays are reported, they tend to be ignored in the subsequent
literature. For instance, in a seminal study of postural balance, Peterka
mentioned that the average overall delay in the control loop was $206\pm 11$
(SD) ms [23]. This figure (approx. 200 ms) was subsequently used as a point
estimate in a number of postural control studies (e.g., [7]), even though the
actual values of the estimates reported in Ref. [23] were scattered in the
range of approx. 100 to 250 ms depending on the subject and task parameters.
Similar delays were found in another experimental paradigm, stick balancing.
Mehta and Schaal [17] reported the average delays of 220 ms in real stick
balancing, and 269 ms in a virtual version of the same task, but the inter-
subject and inter-trial variability suggest that the actual delay values
ranged from approx. 150 to 330 ms. Despite these findings, models of human
intermittent control pervasively continue to employ point estimates of
response delay and ignore its variability.

Besides substantial variability, research on response delay in human
intermittent control is complicated by the notions that neural delays can
possess non-stationary adaptation dynamics [24; 25; 24; 26], and that humans
can employ predictive mechanisms to compensate for response delay [15; 16;
17]. The original formalism of DDE cannot capture any of these features. One
might argue that certain extensions of DDE can capture these properties
individually, for instance, distributed delay [25; 27], stochastic time [28],
time-varying delay [29; 30], predictive feedback [31]. However, to
mathematically describe all these properties together, one would necessarily
need to concatenate multiple corresponding DDE extensions at the same time,
which makes the hypothetical resulting model intractable.

The overarching purpose of the present study is to call for development of
principally new mathematical formalism describing human response delay in all
its complexity. We support our case by the experimentally demonstrating how
complex human response delay is even in a simple balancing task. Specifically,
we directly measure response delay in a virtual stick balancing task
previously used to study the mechanisms of human intermittent control [32].
Our results demonstrate, first, that the response delay is effectively a
random variable which can deviate substantially from its mean value. Second,
when the system dynamics is predictable, the delay can be negative, which
suggests that the subjects sometimes anticipate the system behavior. Third,
the response delay can exhibit oscillatory and adaptive dynamics characterized
by long-range correlations. Based on the current results, we argue that
development of new mathematical formalism is needed to accurately model human
response delay in intermittent control. Such formalism needs to incorporate as
inherent properties 1) the stochastic nature of response delay, 2) its non-
stationary and non-linear dynamics, and 3) possibility of prediction.

##  II Methods

We performed two experiments with two partially overlapping groups of
subjects. The aim of Experiment 1 was to investigate the human response delay
and its dynamics while controlling a system with unpredictable or partially
predictable dynamics. The aim of Experiment 2 was to replicate the results of
Experiment 1, and further investigate the delay dynamics in case the system
dynamics is fully predictable.

The setup of the two experiments was identical, except for two modifications
and an additional condition introduced in Experiment 2. In what follows, we
first describe the general experimental design, and then underline the
distinctions between the two experiments.

###  II.1 Subjects

Eight right-handed healthy male undergraduate students (age 20 to 21)
participated in Experiment 1. Ten right-handed healthy male students (age 19
to 25) participated in Experiment 2. Experimental procedures were approved by
the University of Aizu Ethics Committee. All subjects gave written informed
consent to participate in the experiments.

###  II.2 Stick balancing setup

Virtual overdamped stick balancing was used a representative task involving
human operator control over an unstable system. The experimental setup used
here is based on the virtual balancing task previously used to study the
control activation mechanisms [32; 33]. Here we briefly outline the original
stick balancing task and then describe in details how it was modified to
measure human response delay in the current study.

On the computer screen, an operator observes a vertically oriented stick and a
moving cart connected to the base of the stick via the pivot point (Fig.
1(a)). The task is to maintain the upright position of the stick by moving the
cart horizontally via the computer mouse. The stick motion is simulated by
numerically solving the equation (see Ref. [32] for details)

$\tau_{\theta}\frac{d\theta}{dt}=\sin\theta-\frac{\tau_{\theta}}{l}v\cos\theta\,,$
(1)

where $\theta$ is the angle between the stick and its upright position, and
$v$ is the cart velocity determined by the velocity of the mouse cursor
controlled by the operator. The parameter $\tau_{\theta}$ characterizes the
time scale of stick fall in the absence of subject’s control. The stick length
$l$ determines the characteristic magnitude of the cart’s displacements
required for keeping the stick upright. In the reported experiments we set
$\tau_{\theta}=0.5$ s, which corresponds to the typical time of stick fall
without the operator’s input about 2 s. The stick length was set to 60% of the
screen height (21.5 inch screen was used in the reported experiment).

(a)

(b)

Figure 1: (a) Stick balancing task used in the random and one-side
conditions(Experiments 1 and 2) and visible condition (Experiment 2). (b)
Hypothetical trial dynamics in the visible condition (Experiment 2): stick
angle $\theta$, stick accessibility $\eta$, and mouse velocity $v$.

###  II.3 Response delay measurement

####  II.3.1 Experiment 1

In Experiment 1, we investigated human response delay patterns under two
conditions, random and one-side, each including 30 practice trials and 300
recorded trials. In both conditions, each trial started with the cart
positioned in the middle of the screen and the stick placed inside the
invisibility sector $|\theta_{c}|<5^{\circ}$. The subjects’ goal was to start
controlling the stick as soon as it becomes visible, and return it to the
vicinity of the vertical position (Fig. 1(a)). In the random condition, the
stick was initially deviated to the left or to the right of the vertical
position, with the sign of deviation determined randomly prior to each trial
($\theta_{0}=\pm 0.5^{\circ}$). In the one-side condition, the stick was
always deviated to the right of the vertical position
($\theta_{0}=0.5^{\circ}$). Therefore, in the random condition subjects could
not predict where and when the stick would appear, but in the one-side
condition the subjects knew the side but not the time of the stick appearance.
Each trial continued for 5 s; in case the stick fell down, the trial was
terminated earlier. Each trial was followed by a 3 s rest period. All subjects
completed both random and one-side versions of the task, with the condition
order counterbalanced (half the subjects completed the random condition first,
and the other half completed the one-side condition first).

####  II.3.2 Experiment 2

Experiment 2 included the same random and one-side conditions as Experiment 1,
with two modifications. First, when the stick was in the invisibility region,
the mouse movements did not affect the cart position. This constraint was
introduced to allow for registering predictive responses. Second, each trial
was only terminated when the stick was returned to the invisibility sector
($|\theta|<\theta_{c}$) and, at the same, the cart velocity was sufficiently
close to zero ($|v_{n}|<v_{c}$). This was done to ensure that the subjects
respond to the detected stick deviation with a precise corrective movement,
which would take into account not only the sign, but also the magnitude of the
deviation.

In Experiment 2 we also introduced an additional condition, visible stick. In
this condition, the stick was visible at all times, but the subjects’ mouse
cursor movements did not affect the cart and the stick while the stick was in
the inaccessibility sector $|\theta_{c}|<10^{\circ}$ (Fig. 1(a)). In this way,
the subjects could observe the stick motion and could potentially time their
response to coincide with the moment the stick leaves the inaccessibility
sector.

As there was no evidence of condition order effects in Experiment 1, in
Experiment 2 all subjects performed the experiment in the following order of
conditions: visible, random, one-side. Each subject performed 30 practice and
400 recorded trials per condition.

To prevent small-scale, involuntary mouse movements to affect the stick motion
in the invisibility/inaccessibility sector, we applied a threshold
$v_{c}=0.01$ to mouse velocity during the experiment (the characteristic scale
of mouse velocity fluctuations was about 0.5).

###  II.4 Data analysis

The response delay was defined as the time lag between the moment when the
stick becomes visible/accessible and the moment when a subject starts to move
the computer mouse, i.e., when the mouse velocity exceeds the cutoff value
$v_{c}=0.01$ (Fig. 1(b)). Note that even though in Experiment 2 the cart was
inaccessible until the stick angle exceeded $\theta_{c}$, the subjects could
still start moving the mouse in advance, trying to anticipate the stick
appearance. In such cases, the delay would take negative values.

In Experiment 1, the subjects’ mouse movements could affect the stick motion
even when the stick was still in the invisibility region. For this reason, we
excluded from our analysis the trials where the subjects moved the cart before
the stick became visible, as these movements changed the controlled system
dynamics, rendering the delay measurement impossible. The number of such
trials varied across subjects (1% to 45%), and was on average 12% in the
random condition and 8% in the one-side condition.

##  III Results

### Experiment 1

We hypothesized that the random condition will result in larger response
delays than the one-side condition. In the former, the direction of the
stimulus is unknown, so the subjects’ response presumably involved additional
time needed to make a decision on the response direction. Indeed, in all eight
subjects the response delay in the random condition was on average larger than
in the one-side condition (Tab. 1).

Table 1: Mean delay values, standard deviations, and the results of the D’Agostino’s K-squared normality test for delay times obtained in Experiment 1 Condition | Subject | mean | std |  $K^{2}$ statistic |  $p$-value
---|---|---|---|---|---
Random | 1 | 0.29 | 0.08 | 53 | 3$\times 10^{-12}$
| 2 | 0.37 | 0.08 | 221 | 1$\times 10^{-48}$
| 3 | 0.29 | 0.12 | 9 | 9.44$\times 10^{-4}$
| 4 | 0.31 | 0.10 | 35 | 2.67$\times 10^{-8}$
| 5 | 0.41 | 0.22 | 30 | 2.76$\times 10^{-7}$
| 6 | 0.41 | 0.14 | 116 | 5.86$\times 10^{-26}$
| 7 | 0.39 | 0.13 | 121 | 4.91$\times 10^{-27}$
| 8 | 0.32 | 0.10 | 50 | 1.66$\times 10^{-11}$
One-side | 1 | 0.27 | 0.09 | 13 | 1.91$\times 10^{-3}$
| 2 | 0.26 | 0.14 | 20 | 4.31$\times 10^{-5}$
| 3 | 0.24 | 0.13 | 14 | 8.11$\times 10^{-4}$
| 4 | 0.28 | 0.09 | 17 | 2.23$\times 10^{-4}$
| 5 | 0.26 | 0.14 | 9 | 9.32$\times 10^{-4}$
| 6 | 0.34 | 0.16 | 81 | 2.6$\times 10^{-18}$
| 7 | 0.33 | 0.11 | 58 | 2.93$\times 10^{-13}$
| 8 | 0.27 | 0.10 | 16 | 3.01$\times 10^{-4}$

Analysis of dynamics of the response delay across trials has shown that it
exhibited roughly stationary behavior in both conditions in all subjects (Fig.
2, upper panels). The delay fluctuations were uncorrelated, showing little
evidence for complex dynamics in Experiment 1 (Fig. 2, middle panels).

(a) Random condition

(b) One-side condition

Figure 2: Results of Experiment 1: response delay dynamics, autocorrelation function, and distribution in (a) random and (b) one-side conditions. Table 2: Mean delay values, standard deviations, and the results of the D’Agostino’s K-squared normality test for delay times obtained in Experiment 2 Condition | Delay dynamics type | Subject | mean | std |  $K^{2}$ statistic | p
---|---|---|---|---|---|---
Random | Stationary | 1 | 0.27 | 0.14 | 124.57 | $8.91\times 10^{-28}$
|  | 2 | 0.36 | 0.14 | 236.85 | $3.70\times 10^{-52}$
|  | 3 | 0.35 | 0.1 | 157.88 | $5.20\times 10^{-35}$
|  | 4 | 0.44 | 0.12 | 160.58 | $1.35\times 10^{-35}$
|  | 5 | 0.28 | 0.2 | 101.25 | $1.03\times 10^{-22}$
|  | 6 | 0.29 | 0.19 | 101.19 | $1.06\times 10^{-22}$
|  | 7 | 0.33 | 0.16 | 66.04 | $4.56\times 10^{-15}$
|  | 8 | 0.35 | 0.1 | 334.38 | $2.46\times 10^{-73}$
|  | 9 | 0.28 | 0.13 | 140.14 | $3.71\times 10^{-31}$
|  | 10 | 0.3 | 0.12 | 262.17 | $1.18\times 10^{-57}$
One-side | Stationary | 4 | 0.35 | 0.11 | 123.95 | $1.21\times 10^{-27}$
|  | 6 | 0.14 | 0.22 | 49.87 | $1.48\times 10^{-11}$
|  | 7 | 0.26 | 0.24 | 32.29 | $9.72\times 10^{-08}$
|  | 8 | 0.25 | 0.14 | 110.05 | $1.26\times 10^{-24}$
|  | 9 | 0.17 | 0.18 | 18.83 | $8.13\times 10^{-05}$
|  | 10 | 0.34 | 0.16 | 185.27 | $5.88\times 10^{-41}$
| Oscillations | 1 | -0.44 | 0.39 | 29.17 | $4.64\times 10^{-07}$
|  | 2 | 0.02 | 0.26 | 12.73 | $1.72\times 10^{-03}$
|  | 3 | 0.0 | 0.19 | 12.44 | $1.99\times 10^{-03}$
|  | 5 | -0.29 | 0.32 | 7.22 | $2.71\times 10^{-02}$
Visible | Stationary | 3 | -0.25 | 0.15 | 69.86 | $6.75\times 10^{-16}$
|  | 5 | -0.38 | 0.28 | 14.64 | $6.63\times 10^{-04}$
|  | 8 | -0.05 | 0.11 | 140.28 | $3.46\times 10^{-31}$
|  | 9 | -0.01 | 0.09 | 80.7 | $2.99\times 10^{-18}$
|  | 10 | 0.22 | 0.11 | 92.01 | $1.05\times 10^{-20}$
| Oscillations | 1 | -0.32 | 0.36 | 2.24 | $3.27\times 10^{-01}$
|  | 4 | -0.14 | 0.27 | 7.05 | $2.95\times 10^{-02}$
| Adaptation | 2 | -0.14 | 0.19 | 16.21 | $3.02\times 10^{-04}$
|  | 6 | -0.22 | 0.26 | 10.87 | $4.37\times 10^{-03}$
|  | 7 | 0.02 | 0.19 | 39.66 | $2.45\times 10^{-09}$
Figure 3: Results of Experiment 2 (random condition): response delay dynamics
and autocorrelation function of a representative subject, and the histogram of
delays produced by all subjects.

(a) Stationary dynamics

(b) Oscillations

Figure 4: Results of Experiment 2 (one-side condition): Response delay
dynamics and autocorrelation functions of two representative subjects ((a)
Subject 4 and (b) Subject 1), and the distributions of delays produced by all
subjects. The subjects produced two types of delay dynamics: (a) stationary
and (b) oscillatory.

The key finding of Experiment 1 is that the response delay exhibited
substantial variability (Fig. 2, lower panels). In random condition, in most
trials the delay ranged from approx. 200 ms to 500 ms. In the more predictable
one-side condition, the delay ranged from 0 ms to 500 ms, indicating that the
subjects often anticipated the moment when the stick leaves the invisibility
region and utilized this to minimize the response delay. In both conditions,
the within-subject standard deviation of the response delay was approx. $1/3$
of that subject’s mean delay value. Finally, in all subjects and in both
conditions, the D’Agostino $K^{2}$ test has shown significant evidence that
the delay is not normally distributed (Tab. 1). These findings provide
evidence against the fixed delay approximation implied by the DDE-based models
of intermittent control.

### Experiment 2

The purpose of Experiment 2 was two-fold: first, to replicate the results of
Experiment 1 in a different subject group, and, second, to explore the
properties of response delay in the situations where the dynamics of the
controlled system is predictable. Experiment 2 thus involved an additional
condition, visible stick. In addition, in Experiment 2 we allowed for the
subjects’ response delay to be negative in the random and one-side conditions
by disconnecting the mouse movements from the cart while the stick was in the
invisibility sector. This enabled us to capture the signs of prediction
mechanisms when the subjects controlled a partially predictable system.

The random condition results corroborate the findings of Experiment 1: In the
random condition, the delay dynamics was largely stationary, with uncorrelated
delay fluctuations (Fig. 3), and delay variability was characterized by
standard deviations approx. $1/3$ to $1/2$ of the mean (Table 2). Very few
trials resulted in negative delay (cart movement initiated before the stick
leaves the inaccessibility region), which suggests that predictions mechanisms
were not efficient in the case of unpredictable stick motion direction.

The one-side condition results were also consistent with the corresponding
results of Experiment 1, with two main distinctions. First, many subjects
exploited the possibility of initiating the cart movement in anticipation of
the stick appearance. In six subjects, this resulted in the response delay
being negative in a substantial portion of trials (Fig. 4, lower panels).
Second, we found that four subjects produced non-stationary time patterns of
response delay. Particularly, the wave-like autocorrelation functions of the
delay time series in Subjects 1, 2, 3, and 5 are the signs of periodic
dynamics (Fig. 4(b)). One of the possible explanations for this is that these
subjects regularly switched between two or more response strategies, e.g.,
responding to the stimulus only after it becomes visible vs. trying to predict
the stimulus appearance in advance.

Finally, in the visible condition where the stick motion was fully observable
and thus most predictable among all the conditions, we found an even more
diverse spectrum of response delay dynamics. Similar to the one-side
condition, some subjects employed the strategy of stationary response delay
(Fig. 5(a)), and few subjects produced oscillating delay dynamics (Fig. 5(b)).
However, in addition to these two patterns observed in the one-side condition,
we found that the response delay of three subjects showed the signs of
adaptation. The time patterns of the delay produced by subjects 2, 6, and 7
show gradual decrease over time, which is characterized by long-range self-
correlations. This suggests that these subjects learned the dynamics of the
visible stick over time, and adapted their response correspondingly, which
resulted in smaller delays.

(a) Stationary dynamics

(b) Oscillations

(c) Adaptation

Figure 5: Results of Experiment 2 (visible condition): Response delay dynamics
and autocorrelation functions of three representative subjects ((a) Subject 3,
(b) Subject 1, and (c) Subject 2), and the distributions of delays produced by
all subjects. The subjects produced three types of delay dynamics: (a)
stationary, (b) oscillatory, and (c) adaptive.

##  IV Discussion

In this paper, we demonstrated that

  * •

Human response delay is characterized by substantial individual differences;

  * •

It can be distributed across the wide range of values;

  * •

When the system under control is at least partially predictable, the delay can
be compensated or over-compensated by anticipatory mechanisms;

  * •

The delay can exhibit non-stationary and non-linear dynamics characterized by
long-range correlations.

All these properties were observed together in a simple virtual balancing
task. This suggests that in more sophisticated control processes, human
response delay is at least as complex as in the task considered here. Studies
of postural balance and stick balancing of a fingertip widely acknowledge the
importance of response delay in human control, but almost universally utilize
the fixed-delay approximation provided by delay-differential equations (e.g.,
[1; 34; 7]). Our results imply that this approximation does not provide an
adequate mathematical description of the human response delay.

Previously, uncertainty in human response timing was described in the model of
random delay regarded in terms of random time with fixed delay or the delay
changing randomly with time [28; 35]. The concept of time-varying delay,
$\tau=\tau(t)$, is also met in describing the dynamics of a stem cell
population [36] and neural network behavior [37] including random variations
[e.g., 38]. There are also phenomena in population dynamics whose description
turns to the idea of the delay time $\tau=\tau(x)$ depending on the current
system state $x(t)$ [39]. However, overall the models encompassing the idea of
non-constant time delay are rare, possibly because such models are currently
difficult to study both analytically and numerically.

Intrinsic features of human sensorimotor system require the mathematical
models describing human control to capture not only response delay, but also
sensory deadzones (separately for position, velocity, and acceleration) [40;
41; 42], noise [1; 9], and prediction [17; 31; 3; 43]. Furthermore, such
models may need to take into account the structure of the state space of the
controlled system, e.g., saddle-type instability of the equilibrium in case of
inverted pendulum with inertia [34; 7; 44; 45; 46]. Capturing all these
features together using the standard formalism of differential equations
(stochastic and time-delayed) would inevitably result in a model which is
mathematically intractable. For this reason, we argue that principally new
mathematical formalism should be developed for describing human intermittent
control and similar phenomena.

Finally, we briefly outline a formalism that can potentially allow for the
found features of human control in a natural way. The main drawback of the
approaches to describing human control over unstable systems within the
paradigm of delay differential equations is that this formalism does not take
into account some of the basic features of human perception and cognition.
Namely, the spatial point and the point-like instant of time are the main
elements it deals with. However, we argue that these objects are ill-defined
in the context of human cognition, so their use for describing human response
might be problematic.

Indeed, first, our mental evaluation of a spatial position of a system under
control with respect to its environment can be done only approximately;
therefore, one may argue that in our consciousness instead of spatial points
we operate with some regions with fuzzy boundaries. The concept of dynamical
trap — a certain region where any point is treated by operator as equilibrium
— was developed as an alternative to the notion of stationary point to
describe phenomenologically how humans maintain a system near its unstable
equilibrium [47; 48; 49]. It should be emphasized that the notion of dynamical
trap is not reduced to the introduction of perception threshold because the
control activation is an essentially probabilistic phenomenon [32; 50; 5].
Even a simplified dynamical trap model [e.g., 49] has to combine the elements
of fuzzy control and the theory of stochastic processes to describe such human
actions. Furthermore, dynamical traps can give rise to new type non-
equilibrium phase transitions not meet in physical systems [47; 48; 51; 52].
The notion of dynamical trap is also the pivot point in the novel concept of
noise-induced activation of human intermittent control [32; 50].

Second, the analysis of human temporal experiences also leads to the
conclusion that our internal now is not a point-like instant of time. The now
existing in the mind is represented by a finite time interval whose points are
perceived as simultaneous — the term specious present is widely used for this
human now. There is a vast literature devoted to such temporal experiences and
a reader may be referred, e.g., to [53; 54] for reviews and other references.
For this reason the point-like instant should also replaced by a certain
analogy to dynamical traps in time.

The concept of space-time cloud [55] — a certain fuzzy region on the space-
time continuum playing the role of here-and-now in our consciousness and being
an indivisible in it object — allows for both these features. The real
position of a given system and the current instant of time are hidden
quantities for our mind and we are not able to operate with them directly. In
this case the mental comparison of two objects can be implemented as the
overlap of the two clouds.

In this way we come to a conclusion that the desired formalism of describing
response delays and uncertainty of spatial position should operate with space-
time clouds and their dynamics. In this case all the effects discussed in the
present work — including the probabilistic nature of human response delay, its
wide distribution, the contribution of immediate past and anticipation — can
potentially be allowed for directly. The spatial and temporal fuzziness of a
space-time cloud may be represented in terms of some function $\Psi(x,v,t)$
whose dynamics is governed by the equations dealing with such functions as
points of Hilbert space [a preliminary discussion of this idea can be found in
56]. In this sense the formalism of space-time clouds resembles that of
quantum dynamics, however, it must be distinct from quantum dynamics in
several aspects. One of them is fuzziness with respect to the temporal
dimension caused by the finite thickness of human now. It reflects in that the
persistence and change are experienced directly as mutually irreducible,
complementary aspects of human perception [e.g., 57]. For this reason, in
contrast to quantum mechanics, the function $\Psi(x,v,t)$ contains the spatial
coordinate $x$ together with the velocity $v$ in the list of its argument.
Developing such models is the subject of future research.

All the data and program code produced in this study are available online:
https://osf.io/eswf5/.

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