# On the action of the long cycle on the Kazhdan-Lusztig basis

Fern Gossow1<EMAIL_ADDRESS>Oded Yacobi1<EMAIL_ADDRESS>Oded Yacobi was partially supported by the ARC grant DP180102563 1School of
Mathematics and Statistics, University of Sydney, Australia

###### Abstract

The complex irreducible representations of the symmetric group carry an
important canonical basis called the Kazhdan-Lusztig basis. Although it is
difficult to express how general permutations act on this basis, some
distinguished permutations have beautiful descriptions. In 2010 Rhoades showed
that the long cycle $(1,2,...,n)$ acts by the jeu-de-taquin promotion operator
in the case when the irreducible representation is indexed by a rectangular
partition. We prove a generalisation of this theorem in two directions: on the
one hand we lift the restriction on the shape of the partition, and on the
other hand we enlarge the result to the collection of all separable
permutations.

###### keywords:

Specht modules, Kazhdan-Lusztig basis, promotion, evacuation

##  1 Main Results

The complex irreducible representations, aka Specht modules, of the symmetric
group $S_{n}$ are indexed by partitions of $n$: for such a partition
$\lambda\vdash n$ we denote by $S^{\lambda}$ the associated representation.
These carry a canonical basis called the Kazhdan-Lusztig basis $\\{C_{T}\mid
T\in SYT(\lambda)\\}$, which is indexed by the set of standard Young tableaux
of shape $\lambda$.

Although the Kazhdan-Lusztig basis enjoys many wonderful properties, it is
quite complicated to express its transformation under the action of an
arbitrary permutation. Nevertheless, owing to a close connection with the RSK
correspondence, it is possible for certain distinguished elements.

In the mid 1990s Berenstein-Zelevinsky [1] and Stembridge [10] showed
independently that the long element of $S_{n}$ acts on the Kazhdan-Lusztig
basis (up to sign) by Schützenberger’s evacuation operator. More recently,
Rhoades proved in the case of rectangular partitions that the long cycle
$c=(1,2,\ldots,n)$ acts on the Kazhdan-Lusztig basis (up to sign) by the jeu-
de-taquin promotion operator, $\mathrm{pr}$ [8]. This was the essential new
ingredient needed in Rhoades’ cyclic sieving phenomenon regarding the action
of promotion on rectangular standard Young tableaux.

Our first main result is a generalisation of Rhoades’ Theorem to arbitrary
partitions. Fix a partition $\lambda\vdash n$ and number its removable boxes
$1,2,\dots,r$ moving downward. Define the _index_ of $T\in SYT(\lambda)$ to be
the said number of the removable box containing $n$, and denote it
$\operatorname{idx}(T)$. Choose an ordering of the Kazhdan-Lusztig basis which
is weakly increasing along the index. For $w\in S_{n}$ let $[w]$ be the matrix
of $w:S^{\lambda}\to S^{\lambda}$ in this ordered basis.

######  Theorem 1.1.

Let $\lambda\vdash n$ be an arbitrary partition. Let $[c]=QR$ be the QR
decomposition of $[c]$ into the product of an orthogonal matrix $Q$ and an
upper triangular matrix $R$. Then $Q$ is a generalised permutation matrix for
$\mathrm{pr}$ with entries $\pm 1$, i.e. for any $T\in SYT(\lambda)$ we have

$QC_{T}=\pm C_{\mathrm{pr}(T)}.$

The sign appearing above depends only on the index of $T$.

Equivalently, this theorem states that $c$ acts on the Kazhdan-Lusztig basis
by promotion on the leading term, i.e.

$c\cdot C_{T}=\pm C_{\mathrm{pr}(T)}+\sum_{R}a_{R}C_{\mathrm{pr}(R)},$

where the sum is over $R$ with index strictly smaller than the index of $T$,
and $a_{R}\in\mathbf{Z}$. In the case of a rectangular partition, the sum is
empty and we recover Rhoades’ Theorem.

The key property of rectangular partitions that Rhoades leverages is that the
restriction of the corresponding Specht module to $S_{n-1}$ remains
irreducible, and the Kazhdan-Lusztig basis is unchanged if the space is
regarded as an $S_{n-1}$-module or an $S_{n}$-module. It is therefore not
surprising that the key step in our argument involves the interaction between
the restriction functor and the Kazhdan-Lusztig basis in a more general
setting, which may be interesting in its own right.

For $k\geq 0$ define the subspace $S^{\lambda}_{k}=\mathrm{span}\\{C_{T}\mid
T\in SYT(\lambda)\text{ and }\operatorname{idx}(T)\leq k\\}$ of $S^{\lambda}$.
This gives rise to a filtration $0=S^{\lambda}_{0}\subset
S^{\lambda}_{1}\subset S^{\lambda}_{2}\subset\cdots\subset
S^{\lambda}_{r}=S^{\lambda}$, where $r$ is the number of removable boxes of
$\lambda$.

######  Theorem 1.2.

Each subspace $S_{i}^{\lambda}$ is $S_{n-1}$-invariant, and for every $1\leq
i\leq r$, there is an isomorphism of $S_{n-1}$-modules
$S^{\lambda}_{i}/S^{\lambda}_{i-1}\cong S^{\mu_{i}}$, where $\mu_{i}\vdash
n-1$ is the partition obtained from $\lambda$ by deleting the $i^{\text{th}}$
removable box.

Note that this theorem provides a new proof that the branching law for
symmetric groups is multiplicity-free, as

$\mathrm{res}^{S_{n}}_{S_{n-1}}S^{\lambda}\cong\bigoplus_{i=1}^{r}S_{i}^{\lambda}/S_{i-1}^{\lambda}\cong\bigoplus_{i=1}^{r}S^{\mu_{i}}.$

Finally, we mention a generalisation of Theorem 1.1 to the class of all
_separable permutations_ , which includes the long element and the long cycle.
By definition, separable permutations are those that can be obtained from
$e\in S_{1}$ by repeated applications of direct sum and skew sum.
Alternatively, these are the permutations that avoid the patterns $2413$ and
$3142$ [6].

Fix $\lambda\vdash n$ an arbitrary partition. In Section 5, we describe
briefly how to associate to each separable permutation $w\in S_{n}$ a
bijection $\varphi$ of $SYT(\lambda)$ such that an analogous result to Theorem
1.1 holds. That is, if $[w]:S^{\lambda}\to S^{\lambda}$ is the matrix giving
the action of $w$ on some given ordering of the KL basis, we have:

######  Theorem 1.3.

Let $w\in S_{n}$ be any separable permutation, and let $[w]=QR$ be the QR
decomposition. Then $Q$ is a (generalised) permutation matrix of $\varphi$.

In specific cases, we can prove this combinatorially but the general case
requires results from categorical representation theory and in particular the
action of braid groups on triangulated categories [4]. In our forthcoming work
we will describe this connection, and use this to also derive variations of
these theorems for arbitrary simply-laced semisimple Lie algebras.

##  2 Background

In this section we briefly recall some basic results we’ll need. We refer the
reader to [9] for more details on notions from algebraic combinatorics, and to
[2] for the necessary background on Kazhdan-Lusztig theory.

###  2.1 Combinatorics

Let $\lambda\vdash n$ be a partition of $n$, which we depict using its
associated Young diagram. Recall that a box in $\lambda$ is called a
_removable box_ if its deletion results in another Young diagram. We label the
removable boxes of $\lambda$ by $1,2,\dots,r$ starting from the top, and
moving down the tableau. For example, the partition $(6,6,3,1)\vdash 16$ has
three removable boxes, with labelling given by:

$\young(~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}1,~{}~{}2,3)$

Write $SYT(\lambda)$ for the set of standard Young tableaux of shape
$\lambda$. We define the _index_ of $T\in SYT(\lambda)$, denoted
$\operatorname{idx}(T)$, to be the label of the removable box containing $n$
as above. If $R$ and $T$ have the same index, then their largest box occupies
the same position. When ordering $SYT(\lambda)$ according to index, we can
break ties by comparing the index values of the tableaux after removing their
largest boxes. Repeating this procedure gives the _total index ordering_ on
$SYT(\lambda)$.

Recall that the _descent set_ $D(T)$ of $T$ is the set of $j$ in
$\\{1,2,\dots,n-1\\}$ such that $j+1$ appears strictly below $j$ in $T$.

The _promotion_ map $\mathrm{pr}:SYT(\lambda)\to SYT(\lambda)$ is defined as
follows: replace the box with value $n$ by a dot, and move this dot north-west
by repeatedly swapping it with the box above or to the left. If both boxes
exist, swap with the larger. Once the box has reached the north-west corner,
increment all values in the tableau and replace the dot with 1.

$\young(134,25)\to\young(134,2\bullet)\to\young(1\bullet
4,23)\to\young(\bullet 14,23)\to\young(125,34)$

The _evacuation_ map $\mathrm{ev}_{n}:SYT(\lambda)\to SYT(\lambda)$ is defined
as follows: denote every box in the tableau ‘fixed’ or ‘unfixed’, with all
boxes initially starting unfixed. At each step, perform the inverse of the
promotion map on the unfixed boxes, and fix the largest remaining box. Repeat
until all boxes are fixed. In the example below fixed boxes are in boldface:

$\young(134,25)\to\young(123,4{{\bf 5}})\to\young(12{{\bf 4}},3{{\bf
5}})\to\young(1{{\bf 3}}{{\bf 4}},2{{\bf 5}})\to\young(1{{\bf 3}}{{\bf
4}},{{\bf 2}}{{\bf 5}})\to\young({{\bf 1}}{{\bf 3}}{{\bf 4}},{{\bf 2}}{{\bf
5}})$

Both promotion and evacuation are bijections on $SYT(\lambda)$, and evacuation
is in fact an involution. It will also be useful to define the _partial
evacuation_ maps. For $1\leq k\leq n$, define the involution
$\mathrm{ev}_{k}:SYT(\lambda)\to SYT(\lambda)$ by fixing all but the smallest
$k$ boxes, and performing evacuation on the remaining unfixed tableau.

######  Lemma 2.1.

For every $T\in SYT(\lambda)$, we have
$\mathrm{pr}(T)=\mathrm{ev}_{n}\mathrm{ev}_{n-1}(T)$.

The _RSK correspondence_ [7] gives a bijection between $w\in S_{n}$ and
ordered pairs $(P,Q)$ of tableaux of some shape $\lambda\vdash n$. We will
denote the RSK correspondence by $w\rightsquigarrow(P,Q)$.

###  2.2 Representation theory

The symmetric group $S_{n}$ is generated by the simple transpositions
$s_{j}:=(j,j+1)$ for $1\leq j\leq n-1$, satisfying the braid relations and
$s_{j}^{2}=1$. Given an element $w\in S_{n}$, we write $l(w)$ for the _length_
of $w$ in these generators.

Given permutations $v,w\in S_{n}$, we write $v<w$ in the _Bruhat order_ if $w$
can be obtained from $v$ by successively swapping two elements in the
permutation’s one-line notation, increasing the length of the intermediate
permutations at each swap. In particular, $v<w$ implies $l(v)<l(w)$. Denote by
$D(w)$ the _(left) descent set_ of $w$, which consists of the indices $j$
between $1$ and $n-1$ such that $s_{j}w<w$.

Let $q$ be an indeterminate. The _Hecke algebra (of type A)_ $H_{n}(q)$ is an
algebra over $\mathbf{Z}[q^{\pm 1/2}]$ generated by
$\\{T_{s_{1}},\dots,T_{s_{n-1}}\\}$ satisfying the braid relations and the
quadratic relation $T_{s_{j}}^{2}=q+(q-1)T_{s_{j}}$ for all $1\leq j\leq n-1$.
As a $\mathbf{Z}[q^{\pm 1/2}]$-module, $H_{n}(q)$ has a basis $\\{T_{w}\\}$
indexed by $w\in S_{n}$. The quadratic relation shows that each $T_{s_{j}}$,
and hence each $T_{w}$, is invertible. The _bar involution_
$\overline{(\cdot)}$ of the Hecke algebra is the $\mathbf{Z}$-linear extension
of the maps $q\mapsto q^{-1}$ and $T_{w}\mapsto T_{w^{-1}}^{-1}$.

######  Theorem 2.2 (Kazhdan-Lusztig Basis, [5, Theorem 1.1]).

There is a unique basis $\\{C_{w}(q)\\}$ of $H_{n}(q)$ indexed by $S_{n}$,
with elements of the form

$C_{w}(q)=q^{-l(w)/2}\sum_{v\in
S_{n}}(-1)^{l(v)-l(w)}q^{-l(v)}P_{v,w}(q)T_{v},$

such that:

  * •

$P_{v,w}(q)$ is an integer polynomial, and is non-zero only if $v\leq w$,

  * •

$P_{v,v}(q)=1$ for all $v\in S_{n}$,

  * •

if $v<w$ then $deg(P_{v,w}(q))\leq(1/2)(l(w)-l(v)-1)$, and

  * •

$\overline{C_{w}(q)}=C_{w}(q)$ for all $w\in S_{n}$.

The $P_{v,w}(q)$ are the well-known _Kazhdan-Lusztig (KL) polynomials_. The
degree bound gives us a statistic on pairs of permutations. For $v\leq w$, we
write $\overline{\mu}(v,w)\in\mathbf{Z}$ for the coefficient of degree
$(1/2)(l(w)-l(v)-1)$ in $P_{v,w}(q)$. If $l(w)-l(v)$ is even or $v$ is
incomparable with $w$ in the Bruhat order, this coefficient will be zero. We
also define $\overline{\mu}(w,v):=\overline{\mu}(v,w)$ when $w<v$, so
$\overline{\mu}$ is symmetric.

Under the specialisation $q=1$, we have the relation $T_{s_{i}}^{2}=1$, and so
$H_{n}(1)\cong\mathbf{Z}S_{n}$. Set $C_{w}=C_{w}(1)\otimes
1\in\mathbf{Z}S_{n}\otimes_{\mathbf{Z}}\mathbf{C}=\mathbf{C}S_{n}$, so that
$\\{C_{w}\mid w\in S_{n}\\}$ is a basis for $\mathbf{C}S_{n}$. Define a binary
relation on $S_{n}$ by setting $v\leq_{L}w$ if, for some $j$, $C_{v}$ appears
with nonzero coefficient in the expansion of $s_{j}\cdot C_{w}$. Taking the
transitive closure gives a preorder on $S_{n}$, called the _left KL preorder_.
By its construction, the span of $\\{C_{v}\mid v\leq_{L}w\\}$ for a fixed
$w\in S_{n}$ will be a submodule of $\mathbf{C}S_{n}$. The equivalence classes
induced by $\leq_{L}$ are the _left cells_ of $S_{n}$, and we write
$v\sim_{L}w$ if $v$ and $w$ belong to the same left cell.

For a left cell $\mathcal{C}$ of $S_{n}$, we write $S^{\mathcal{C}}$ for the
linear span of elements $\\{C_{v}\mid v\in\mathcal{C}\\}$. The following
defines an irreducible action of $S_{n}$ on $S^{\mathcal{C}}$ [2, (6.4)]:

$\displaystyle s_{j}\cdot C_{v}=\begin{cases}-C_{v}&j\in D(v),\\\
C_{v}+\sum_{w\in\mathcal{C},j\in D(w)}\overline{\mu}(v,w)C_{w}&j\not\in
D(v).\end{cases}$ (2.3)

Left cells in $S_{n}$ are determined by the $Q$-tableau in the RSK
correspondence. That is, $v\sim_{L}w$ if and only if $v\rightsquigarrow(P,Q)$
and $w\rightsquigarrow(P^{\prime},Q)$ for some $P,P^{\prime},Q\in
SYT(\lambda)$ and $\lambda\vdash n$. Therefore we can uniquely associate each
left cell $\mathcal{C}$ of $S_{n}$ to a tableau $Q$ with $n$ boxes, and write
$\mathcal{C}_{Q}$. Appropriately restricting the RSK correspondence gives a
bijection between the basis elements $C_{v}\in\mathcal{C}_{Q}$ and $T\in
SYT(\lambda)$, by identifying $C_{v}$ with the $P$-tableau of $v$. Hence, we
can reindex the basis of $S^{\mathcal{C}_{Q}}$ as $\\{C_{T}\mid T\in
SYT(\lambda)\\}$. We can rewrite the action on the KL basis $\\{C_{T}\\}$ in
terms of tableaux only:

  1. 1.

Let $v\in S_{n}$ and suppose $v\rightsquigarrow(P,Q)$. Then $j\in D(v)$ if and
only if $j\in D(P)$.

  2. 2.

For $\lambda\vdash n$ and tableaux $P,T,Q,Q^{\prime}\in SYT(\lambda)$ we have:

$\overline{\mu}((P,Q),(T,Q))=\overline{\mu}((P,Q^{\prime}),(T,Q^{\prime})),$

where we identify permutations with their images under the RSK correspondence.

From this we obtain that for tableaux $Q,Q^{\prime}\in SYT(\lambda)$, the
respective representations $S^{\mathcal{C}_{Q}}$ and
$S^{\mathcal{C}_{Q^{\prime}}}$ are equal (not just isomorphic) up to a
reindexing of basis elements.

These results have a number of implications. First, we can define the
$\overline{\mu}$ value between tableaux $P,T\in SYT(\lambda)$ by
$\overline{\mu}(P,T)=\overline{\mu}((P,Q),(T,Q))$ for any $Q\in SYT(\lambda)$.
Furthermore, since $S^{\mathcal{C}_{Q}}$ depends only on the shape of $Q$, we
can write $S^{\lambda}$ for the representation $S^{\mathcal{C}_{Q}}$, where
$Q$ is any tableau of shape $\lambda$. Finally, we can write the conditions on
the descent set of $v$ in (2.3) in terms of the $P$-tableau of $v$ under RSK.
We incorporate these results into the following theorem.

######  Theorem 2.4 ([2, Theorem 6.5.3]).

Let $\lambda\vdash n$ be a partition. The module $S^{\lambda}$ is irreducible,
and has a basis $\\{C_{T}\mid T\in SYT(\lambda)\\}$ with the following action:

$\displaystyle s_{j}\cdot C_{T}=\begin{cases}-C_{T}&j\in D(T),\\\
C_{T}+\sum_{R\in SYT(\lambda),j\in D(R)}\overline{\mu}(T,R)C_{R}&j\not\in
D(T).\end{cases}$ (2.5)

The irreducible representations of $S_{n}$ are called Specht modules. The
above theorem provides a canonical basis for Specht modules, which we refer to
as the _KL basis_.

##  3 The Branching Rule via the KL Basis

In this section we prove Theorem 1.2. Fix $\lambda\vdash n$, and recall the
filtration of $S^{\lambda}$ by the subrepresentations $S_{i}^{\lambda}$, each
defined as the subspace spanned by $\\{C_{T}\\}$ with
$\operatorname{idx}(T)\leq i$. We will first show that $S_{i}^{\lambda}$
always remains invariant under $S_{n-1}$.

Let $CSS(\lambda)$ be the _column super-strict_ tableau of shape $\lambda$,
where the values $1,2,\dots,n$ are placed column-by-column, reading left-to-
right. We take $\lambda=(4,3,1)$ as an example:

$CSS(\lambda)=\young(1468,257,3)$

Preimages of the RSK correspondence when $Q=CSS(\lambda)$ are easy to
describe. We simply write the elements of the $P$-tableau in column order,
reading from bottom-to-top.

$85162734\quad\rightsquigarrow\quad\left(\,\young(1234,567,8),\,\young(1468,257,3)\,\right)$

######  Lemma 3.1.

Fix $\lambda\vdash n$ and $1\leq i\leq r$, where $r$ is the number of
removable boxes of $\lambda$. Then $S_{i}^{\lambda}$ is an $S_{n-1}$-submodule
of $S^{\lambda}$.

###### Proof.

If $i=r$, there is nothing to prove. Otherwise, choose arbitrary tableau $T,R$
in $SYT(\lambda)$ with $C_{T}\in S_{i}^{\lambda}$ and $C_{R}\not\in
S_{i}^{\lambda}$, or equivalently, $\operatorname{idx}(T)\leq
i<\operatorname{idx}(R)$. Instead of working in $S^{\lambda}$, we work in the
module $S^{\mathcal{C}_{CSS(\lambda)}}$. To this end, define the unique
permutations $v,w\in S_{n}$ with images $(T,CSS(\lambda))$ and
$(R,CSS(\lambda))$ under RSK. We are required to show that $C_{w}$ does not
appear in $s_{j}\cdot C_{v}$ (see (2.5) for the action) when $1\leq j\leq
n-2$. We consider cases depending on the Bruhat comparibility of $v$ and $w$.

(a) $v$ and $w$ are not Bruhat compatible. Then $\overline{\mu}(v,w)=0$ by
definition.

(b) $w<v$. We have a chain $w=w_{1}\to\cdots\to w_{k}=v$ of swaps with
$l(w_{m})<l(w_{m+1})$. At some point, the swap $w_{m}\to w_{m+1}$ moves $n$ to
the right. But this will decrease the length, which is a contradiction.

(c) $v<w$ and $l(v,w)>1$. Suppose $C_{w}$ appears in the expansion of
$s_{j}\cdot C_{v}$. Then $j\in D(w)$ and $j\not\in D(v)$, so
$\overline{\mu}(v,w)=0$ by [2, Proposition 5.1.9].111Björner [2] uses the
convention of right descent sets, but the result for left descent sets is
analogous.

(d) $v<w$ and $l(v,w)=1$. Again, suppose $C_{w}$ appears in the expansion of
$s_{j}\cdot C_{v}$ for some $s_{j}\in S_{n-1}$. Then $j\in D(w)$, $j\not\in
D(v)$ and $l(v,w)=1$, which implies $w=s_{j}v$. But $n$ occurs at different
positions in the one-line notation of $v$ and $w$, so we must have $j=n-1$.
This is again a contradiction, since $s_{n-1}\not\in S_{n-1}$. ∎

Recall that $\mu_{i}$ is the partition obtained by deleting the
$i^{\text{th}}$ removable box from $\lambda\vdash n$. There is a bijection
$d_{i}:\\{T\in SYT(\lambda)\mid\operatorname{idx}(T)=i\\}\to SYT(\mu_{i})$
which removes the largest box of each tableau $T$. By taking its linear
extension, we have an isomorphism of vector spaces
$d_{i}:S_{i}^{\lambda}/S_{i-1}^{\lambda}\to S^{\mu_{i}}$ for each $1\leq i\leq
n-1$. Our goal is to show that this is in fact an isomorphism of
$S_{n-1}$-modules. For this, we need to show that the deletion map preserves
the function $\overline{\mu}$.

We again define a specific recording tableau and work with the permutations
explicitly. For a partition $\lambda\vdash n$ with $r$ removable boxes and
$1\leq i\leq r$, define the tableau $CSS_{i}(\lambda)$ by the following
procedure:

  * •

Create a tableau of shape $\lambda$, and place the value $n$ in removable box
$i$.

  * •

Suppose $n$ is placed in the $k^{\text{th}}$ row. For $1\leq m\leq k-1$, place
the value $n-m$ at the end of row $k-m$.

  * •

Place the remaining values $1,\dots,n-k$ into the tableau column by column.

For example, if $\lambda=(4,3,1)$ and $i=2$ we have:

$CSS_{i}(\lambda)=\young(1467,258,3)$

The proof that the $\overline{\mu}$ function is preserved under the deletion
map was originally given by Rhoades in the case of rectangular partitions. The
following Lemma appears implicitly in the proof:

######  Lemma 3.2 ([8], Lemma 3.1).

Suppose $u=u_{1}\cdots u_{n-1}$ and $v=v_{1}\cdots v_{n-1}$ are elements of
$S_{n-1}$, written in their one-line notation. For a fixed $0\leq k\leq n-1$,
define the permutations

$\displaystyle u^{\prime}$ $\displaystyle=u_{1}\cdots u_{k}nu_{k+1}\cdots
u_{n-1},$ $\displaystyle v^{\prime}$ $\displaystyle=v_{1}\cdots
v_{k}nv_{k+1}\cdots v_{n-1}$

in $S_{n}$. Then we have an equality of $\overline{\mu}$ values
$\overline{\mu}(u,v)=\overline{\mu}(u^{\prime},v^{\prime})$.

######  Proposition 3.3.

Fix $\lambda\vdash n$, and choose tableaux $T,R\in SYT(\lambda)$, both with
index $i$. Then

$\overline{\mu}(T,R)=\overline{\mu}(d_{i}(T),d_{i}(R))$

###### Proof.

Suppose the removable box $i$ of $\lambda$ occurs in row $k$. Define the
following permutations by their images under the RSK correspondence, using the
tableau $CSS_{i}(\lambda)$.

$\begin{matrix}u\rightsquigarrow(T,CSS_{i}(\lambda))&\quad&\widetilde{u}\rightsquigarrow(d_{i}(T),d_{i}(CSS_{i}(\lambda)))\\\
v\rightsquigarrow(R,CSS_{i}(\lambda))&\quad&\widetilde{v}\rightsquigarrow(d_{i}(R),d_{i}(CSS_{i}(\lambda)))\end{matrix}$

By the above Lemma, it remains to show that
$u=\widetilde{u}_{1}\cdots\widetilde{u}_{n-k}n\widetilde{u}_{n-k+1}\cdots\widetilde{u}_{n-1}$
and the result for $v$ follows _mutatis mutandis_.

The box containing the value $n-k+1$ in $CSS_{i}(\lambda)$ will be in the top
row. Hence, $\widetilde{u}_{n-k+1}$ is inserted directly into the top row when
performing RSK on $\widetilde{u}$. In $u$, we instead insert $n$, which will
also insert into the top row. Inserting $\widetilde{u}_{n-k+1}$ will then push
$n$ into the next row. Each following insertion for $u$ mimics
$\widetilde{u}$, except the $n$ box is constantly pushed down the right-hand
side of the insertion tableau. One can verify that this gives the correct
tableaux-pair for $u$, with an additional $n$-box in row $k$ for each. ∎

For a tableau $T\in SYT(\lambda)$ with index $i$, write $[C_{T}]$ for the
basis element $C_{T}+S_{i-1}^{\lambda}$ of
$S_{i}^{\lambda}/S_{i-1}^{\lambda}$, where $1\leq i\leq n-1$ is the unique
value such that $C_{T}\in S_{i}^{\lambda}$, but $C_{T}\notin
S_{i-1}^{\lambda}$.

By (2.5), for a tableau $T$ with index $i$ we have:

$s_{j}\cdot[C_{T}]=\begin{cases}-[C_{T}]&j\in D(T)\\\
[C_{T}]+\sum_{R}\overline{\mu}(T,R)[C_{R}]&j\not\in D(T)\end{cases}$

In the second case, we sum over $R\in SYT(\lambda)$ with $j\in D(R)$ and
$\operatorname{idx}(R)=\operatorname{idx}(T)$. Then:

$d_{i}s_{j}\cdot[C_{T}]=\begin{cases}-C_{d_{i}(T)}&j\in D(T)\\\
C_{d_{i}(T)}+\sum_{R}\overline{\mu}(T,R)C_{d_{i}(R)}&j\not\in D(T)\end{cases}$

with the sum in the second case as before. If $1\leq j\leq n-2$, then $j\in
D(T)$ if and only if $j\in D(d_{i}(T))$, and likewise for $R$. Hence:

$d_{i}s_{j}\cdot[C_{T}]=\begin{cases}-C_{d_{i}(T)}&j\in D(d_{i}(T))\\\
C_{d_{i}(T)}+\sum_{R}\overline{\mu}(d_{i}(T),d_{i}(R))C_{d_{i}(R)}&j\not\in
D(d_{i}(T))\end{cases}$

This is the action of $s_{j}$ on $C_{d_{i}(T)}$ in $S^{\mu_{i}}$. Thus,
$s_{j}$ and $d_{i}$ commute on every basis element $[C_{T}]$ of
$S_{i}^{\lambda}/S_{i-1}^{\lambda}$ for every generator $s_{j}$ of $S_{n-1}$.
This completes the proof of Theorem 1.2.

##  4 The long cycle action on the KL basis

In this section we prove Theorem 1.1. Let $w_{n}\in S_{n}$ be the long
element, and $w_{n-1}$ be the image of the long element of $S_{n-1}$ under the
standard embedding into $S_{n}$. Our goal is to relate the identity
$c=w_{n}w_{n-1}$ to $\mathrm{pr}=\mathrm{ev}_{n}\mathrm{ev}_{n-1}$ from Lemma
2.1.

Note that $w_{n-1}\in S_{n-1}$ will act on a tableau of size $n-1$ by
$\mathrm{ev}_{n-1}$ (up to sign) [10]. Moreover, $d_{i}$ commutes with
$w_{n-1}$ on $S_{i}^{\lambda}/S_{i-1}^{\lambda}$. Therefore

$\displaystyle w_{n-1}\cdot[C_{T}]$
$\displaystyle=d_{i}^{-1}w_{n-1}d_{i}\cdot[C_{T}]$
$\displaystyle=d_{i}^{-1}w_{n-1}C_{d_{i}(T)}$ $\displaystyle=\pm
d_{i}^{-1}C_{\mathrm{ev}_{n-1}d_{i}(T)}=\pm[C_{\mathrm{ev}_{n-1}(T)}].$

Note that the sign depends only on the shape of $d_{i}(T)$. Since the index of
$T$ matches the index of $\mathrm{ev}_{n-1}(T)$, we obtain:

######  Lemma 4.1.

Fix a partition $\lambda\vdash n$. The action of $w_{n-1}$ on the KL basis
$\\{C_{T}\\}$ of $S^{\lambda}$ is

$w_{n-1}\cdot C_{T}=\pm C_{\mathrm{ev}_{n-1}(T)}+\sum_{R}a_{R}C_{R},$

with the sum over $R\in SYT(\lambda)$ satisfying
$\operatorname{idx}(R)<\operatorname{idx}(T)$, and $a_{R}\in\mathbf{Z}$ are
unknown coefficients. Furthermore, the sign of $C_{\mathrm{ev}_{n-1}(T)}$
depends only on $\operatorname{idx}(T)$.

Theorem 1.1 follows immediately from:

######  Theorem 4.2.

Fix a partition $\lambda\vdash n$. The action of $c$ on the KL basis
$\\{C_{T}\\}$ of $S^{\lambda}$ is

$c\cdot C_{T}=\pm C_{\mathrm{pr}(T)}+\sum_{R}b_{R}C_{\mathrm{pr}(R)},$

with the sum over $R\in SYT(\lambda)$ satisfying
$\operatorname{idx}(R)<\operatorname{idx}(T)$, and $b_{R}\in\mathbf{Z}$ are
unknown coefficients. Furthermore, the sign of $C_{\mathrm{pr}(T)}$ depends
only on $\operatorname{idx}(T)$.

###### Proof.

Reindexing the sum from the Lemma with $R\mapsto{\mathrm{ev}_{n-1}(R)}$,

$w_{n-1}\cdot C_{T}=\pm
C_{\mathrm{ev}_{n-1}(T)}+\sum_{R}a_{\mathrm{ev}_{n-1}(R)}C_{\mathrm{ev}_{n-1}(R)}.$

We set $b_{R}:=a_{\mathrm{ev}_{n-1}(R)}$ and apply the long element to both
sides:

$w_{n}w_{n-1}\cdot C_{T}=\pm C_{\mathrm{ev}_{n}\mathrm{ev}_{n-1}}+\sum_{R}\pm
b_{R}C_{\mathrm{ev}_{n}\mathrm{ev}_{n-1}(R)}.$

Replacing $w_{n}w_{n-1}$ with $c$ and $\mathrm{ev}_{n}\mathrm{ev}_{n-1}$ with
$\mathrm{pr}$ gives the result required. ∎

######  Example 4.3.

Fix $\lambda=(3,1,1)$. We arrange $SYT(\lambda)$ using the total index
ordering:

$\displaystyle\young(145,2,3)\prec\;\young(135,2,4)\prec\;\young(125,3,4)\prec\;\young(134,2,5)\prec\;\young(124,3,5)\prec\;\young(123,4,5)$

Using MAGMA [3] we compute $[c]$ with respect to this ordered basis and
calculate its QR decomposition:

$[c]=\text{\footnotesize$\begin{pmatrix}0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
1&0&0&-1&1&0\\\ 0&0&0&0&0&1\\\ 0&1&0&-1&0&1\\\
0&0&1&0&-1&1\end{pmatrix}$}=\text{\footnotesize$\begin{pmatrix}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 1&0&0&0&0&0\\\ 0&0&0&0&0&1\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\end{pmatrix}\begin{pmatrix}1&0&0&-1&1&0\\\ 0&1&0&-1&0&1\\\
0&0&1&0&-1&1\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\end{pmatrix}$}$

One can verify that $Q$ is precisely the permutation matrix $[\mathrm{pr}]$.

######  Corollary 4.4.

Let $T\in SYT(\lambda)$ and suppose $T$ has index $1$. Then $c\cdot C_{T}=\pm
C_{\mathrm{pr}(T)}$, with the sign depending only on $\lambda$. In particular,
when $\lambda$ is a rectangular partition, then $c$ acts by the promotion
operator on the KL basis of $S^{\lambda}$ up to sign.

##  5 Other permutations acting on the KL basis

In this section we discuss Theorem 1.3. Details will appear in forthcoming
work.

Fix a partition $\lambda\vdash n$. Let $I$ denote the set of generators
$\\{s_{1},\dots,s_{n-1}\\}$ of $S_{n}$. For a non-empty subset $J\subseteq I$,
let $S_{J}\leq S_{n}$ denote the associated parabolic subgroup, and $w_{J}$
its longest element. To each such subset, we can associate a filtration
$0=S_{0}^{\lambda}\subset S_{1}^{\lambda}\subset\cdots\subset
S_{p}^{\lambda}=S^{\lambda}$ of the Specht module such that each subspace
$S_{j}^{\lambda}$ is $S_{J}$-invariant and spanned by KL basis elements, and
the quotient representation $S_{j}^{\lambda}/S_{j-1}^{\lambda}$ is isomorphic
to a simple $S_{J}$-module.

This filtration induces a total preorder $\prec_{J}$ on $SYT(\lambda)$, with
$T\prec_{J}R$ when $C_{T}$ appears strictly before $C_{R}$ in this filtration.
Using partial evacuation operators, we also associate a bijection
$\varphi_{J}$ on $SYT(\lambda)$ such that the action of $w_{J}$ on the KL
basis is given by

$w_{J}\cdot C_{T}=\pm
C_{\varphi_{J}(T)}+\sum_{R\prec_{J}T}a_{R}C_{\varphi_{J}(R)}$

for some constants $a_{R}\in\mathbf{Z}$, and a sign which depends only on the
equivalence class of $T$ under $\prec_{J}$.

We can extend this result in a natural way to _chains_ of subsets of $J$. Call
a permutation _descending_ if it is of the form
$w_{J_{\bullet}}=w_{J_{k}}\cdots w_{J_{1}}$ for some increasing chain
$J_{\bullet}$ of subsets $J_{1}\subset\cdots\subset J_{k}\subseteq J$. It
turns out that a permutation is descending if and only if it is
separable.222This observation was communicated to the authors by Joel Gibson,
who discovered this fact by enumerating the descending permutations
computationally in MAGMA. Each descending permutation has an associated
bijection $\varphi_{J_{\bullet}}=\varphi_{J_{k}}\cdots\varphi_{J_{1}}$ and a
total preorder $\prec_{J_{\bullet}}$ on $SYT(\lambda)$. The statement is as
before, with the action of $w_{J_{\bullet}}$ on the KL basis of $S^{\lambda}$
given by

$w_{J_{\bullet}}\cdot C_{T}=\pm
C_{\varphi_{J_{\bullet}}(T)}+\sum_{R\prec_{J_{\bullet}}T}b_{R}C_{\varphi_{J_{\bullet}}(R)}$
(5.1)

for some constants $b_{R}\in\mathbf{Z}$, and a sign which depends only on the
equivalence class of $T$ under $\prec_{J_{\bullet}}$. Theorem 1.3 follows from
this formula.

When $J\subseteq I$ is a _connected_ subset $\\{s_{a+1},\dots,s_{b}\\}$ for
$0\leq a\leq b<n$, then $S_{J}$ is isomorphic to the symmetric group
$S_{b-a}$. Moreover, $w_{J}=w_{b+1}w_{b-a}w_{b+1}$, where $w_{k}\in S_{n}$ is
the permutation reversing $\\{1,\dots,k\\}$. Likewise,
$\varphi_{J}=\mathrm{ev}_{b+1}\mathrm{ev}_{b-a}\mathrm{ev}_{b+1}$, where
$\mathrm{ev}_{k}$ is the partial Schützenberger involution from Section 2.1.
Finally, the preorder $\prec_{J_{i}}$ can be defined explicitly by:
$R\prec_{J}T$ if and only if $\mathrm{ev}_{a}\mathrm{ev}_{n}(R)$ precedes
$\mathrm{ev}_{a}\mathrm{ev}_{n}(T)$ in the total index ordering.

In the case of chains of _connected_ subsets of $J$, equation (5.1) can be
proved combinatorially. However, in the case of general chains, we use results
from categorical representation theory, which will appear in later work by the
authors. This will also prove analogues of these results in other types
concerning the action of an ADE Weyl group on the Lusztig’s dual canonical
basis in the zero weight space of an irreducible representation of the
corresponding Lie algebra.

Finally, we note that the result in (5.1) does not hold for arbitrary
permutations. For example, take the non-separable permutation $w=2413\in
S_{4}$ and $\lambda=(3,1)$. Choose any of the $3!$ orderings of the KL basis
for $S^{\lambda}$, and compute the $QR$ decomposition of $[w]:S^{\lambda}\to
S^{\lambda}$ with respect to this basis. One can verify that $Q$ is not a
generalised permutation matrix under any of these orderings.

###### Acknowledgements.

The results in Sections 1-4 are part of the first author’s University of
Sydney Honours Thesis (2021), supervised by the second author. The authors
thank the anonymous referees for helpful comments, and Joel Gibson, Ed Heng,
Tony Licata and Eloise Little for useful conversations that improved this
work.

## References

  * [1] A. Berenstein and A. Zelevinsky  “Canonical bases for the quantum group of type $A_{r}$ and piecewise-linear combinatorics”  In _Duke Math. J._ 82.3, 1996, pp. 473–502  DOI: 10.1215/S0012-7094-96-08221-6
  * [2] A. Bjorner and F. Brenti  “Combinatorics of Coxeter Groups”, Graduate Texts in Mathematics  Springer Berlin Heidelberg, 2006  URL: https://books.google.com.au/books?id=1TBPz5sd8m0C
  * [3] W. Bosma, J. Cannon and C. Playoust  “The Magma algebra system. I. The user language” Computational algebra and number theory (London, 1993)  In _J. Symbolic Comput._ 24.3-4, 1997, pp. 235–265  DOI: 10.1006/jsco.1996.0125
  * [4] I. Halacheva, T. Licata, I. Losev and O. Yacobi  “Categorical braid group actions and cactus groups”, 2021  arXiv:2101.05931
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  * [7] D. E. Knuth  “Permutations, matrices, and generalized Young tableaux”  In _Pacific J. Math._ 34, 1970, pp. 709–727  URL: http://projecteuclid.org.ezproxy.library.sydney.edu.au/euclid.pjm/1102971948
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# Static Charged Black Hole Solutions in Hořava-Lifshitz Gravity

Jin-Zhang Tang111Electronic<EMAIL_ADDRESS>Department of
Physics, and State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, China

(August 27, 2024
)

###### Abstract

In the present work, we search static charged black hole solutions to Hořava-
Lifshitz gravity with or without projectability condition. We consider the
most general form of action which electromagnetic field couples with Hořava-
Lifshitz gravity. With the projectability condition, we find (A)dS-Reissner-
Nordstrom black hole solution in Painlevé-Gullstrand type coordinates in the
IR region and a de-Sitter space-time solution in the UV region. Without the
projectability condition, in the IR region, we find an especial static charged
black hole solution.

###### pacs:

98.80.Cq

##  I introduction

The Hořava-Lifshitz gravity which was introduced in Horava:2008ih ;
Horava:2009uw  is intended to be a power-counting renormalizable gravity
theory. The basic idea behind Hořava’s theory is that time and space may have
different dynamical scaling in UV limit. This was inspired by the development
in quantum critical phenomena in condensed matter physics, with the typical
model being Lifshitz scalar field theoryLifshitz ; Chen:2009ka . In this
Hořava-Lifshitz theory, time and space will take different scaling behavior as

$\mathbf{x}\rightarrow b\mathbf{x},\;\;\;\;t\rightarrow b^{z}t,$ (1)

where $z$ is the dynamical critical exponent characterizing the anisotropy
between space and time. Due to the anisotropy, instead of diffeomorphism, we
have the so-called foliation-preserving diffeomorphism. The transformation is
now just

$\displaystyle t$ $\displaystyle\rightarrow$ $\displaystyle\tilde{t}(t),$
$\displaystyle x^{i}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{x^{i}}(x^{j},t).$ (2)

As a result, there is one more dynamical degree of freedom in Hořava-Lifshitz-
like gravity than in the usual general relativity. Such a degree of freedom
could play important role in UV physics, especially in early
cosmologyCai:2009dx ; Chen:2009jr . At IR, due to the emergence of new gauge
symmetry, this degree of freedom is not dynamical any more such that the
kinetic part of the theory recovers the one of the general relativity.

Since time direction plays a privileged role in the whole construction, it is
more convenient to work with ADM metric

$ds^{2}=-N^{2}dt^{2}+h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (3)

in which $N$ and $N_{i}$ are called “lapse” and “shift” variables
respectively.

Taking Hořava-Lifshitz gravity as a new gravitational theory, it is an
important issue to study its black hole solutions. In papers Lu2009 ;
Nastase2009 ; Kehagias:2009is ; AhmadGhodsi2009 ; Colgain:2009fe ; park2009 ;
RongCai-2009 ; Hatefi2009 , it was assumed that the metric of the black
solutions had the following Schwarzschild coordinates form

$ds^{2}=-N(r)^{2}dt_{S}^{2}+\frac{dr^{2}}{g(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (4)

From this metric ansatz, it was found that there were new black hole
solutions, even at IR. For example, in Kehagias:2009is , based on a modified
Hořava-Lifshitz -type action, an asymptotically flat solution with

$g=N^{2}=1+\omega r^{2}-\sqrt{r(\omega^{2}r^{3}+4\omega M)}$ (5)

was found. And paper RongCai-2009  find a new static charged black hole
solution in the IR region, paper Hatefi2009  had studied the extremal rotating
no-charged black hole solutions. However, in the above ansatz (4) the “lapse
function” $N(r)$ obviously breaks the “projectability condition” which means
that $N$ only is the function of $t$. This is introduced in Horava:2008ih ;
Horava:2009uw . For the metric of the form (4), we can work in the Painlevé-
Gullstrand coordinates by making a transformation

$dt_{S}=dt_{PG}-\frac{\sqrt{1-N^{2}}}{N^{2}}dr.$ (6)

Then the ansatz (4) becomes

$ds^{2}=-dt_{PG}^{2}+(dr+\sqrt{1-N^{2}}dt_{PG})^{2}+(\frac{1}{g}-\frac{1}{N^{2}})dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (7)

Comparing with the ADM metric, we find that $N(t_{PG})=1$ which is in accord
with the projectability condition and if $g=N^{2}$ we reach (3). For instance,
a Reissner-Nordstrom black hole after the transform (6) has the form

$ds^{2}=-dt_{PG}^{2}+(dr\pm\sqrt{\frac{2GM}{r}-\frac{Q^{2}}{r^{2}}}dt_{PG})^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (8)

So paper static_black_hole_of_HL  search non-charge black hole solutions to
modified Hořava-Lifshitz gravity which was introduced by Visser_2009  with the
metric ansatz as

$ds^{2}=-N^{2}dt^{2}+\frac{1}{f(r)}(dr+N^{r}dt)(dr+N^{r}dt)+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (9)

where $N$ only is a function of $t$. Paper static_black_hole_of_HL  found
maximally symmetric space solution with curvature $\Lambda_{W}$ and in the IR
region, and (A)dS-Schwarzschild black hole solution in Painlevé-Gullstrand
type coordinates was found. In the UV region, it found a de-Sitter space-time
solution.

Then papers Kiritsis_blackhole ; Capasso_blackhole  search black hole
solutions with the same metric ansatz as (9), but $N$ only is the function of
$r$. They seem get some new black hole solutions. Actually the solutions of
paper Capasso_blackhole  in the IR region is very similar as paper park2009 .
Furthermore, paper Setare2009  had studied the plane symmetric solutions in
Hořava-Lifshitz theory.

In the present work, we search static charged black hole solutions to Hořava-
Lifshitz gravity coupling with electromagnetic field. We work on the same
metric ansatz as (9). Whether the metric ansatz respects the projectability
condition is determined on whether $N$ only is the function of $t$. With the
projectability condition, we find (A)dS-Reissner-Nordstrom black hole solution
in Painlevé-Gullstrand type coordinates in the IR region and a de-Sitter
space-time solution in the UV region. Without the projectability condition, in
the IR region, we find a static charged black hole solution which is similar
as the results of RongCai-2009 . In the UV region, we find the same solution
as Lu2009 .

##  II The Hořava-Lifshitz gravity

In this section, we give a brief review of Hořava-Lifshitz gravity. Using the
ADM formalism, the action of this Hořava-Lifshitz gravitational theory is
given byHorava:2008ih ; Horava:2009uw

$\displaystyle S_{HL}$ $\displaystyle=$ $\displaystyle\int
dtd^{3}\mathbf{x}(\mathcal{L}_{K}+\mathcal{L}_{V}),$
$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})\right\\},$ $\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda^{2}_{W})}{8(1-3\lambda)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}\right.$
(10)
$\displaystyle\left.-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\},$

where $\mathcal{L}_{K}$ is the kinetic term and $\mathcal{L}_{V}$ is the
potential term. If we consider the term which represents a “soft” violation of
the “detailed balance” condition in Horava:2009uw  and add the term in the
action, The kinetic term $\mathcal{L}_{K}$ is the same, and the potential term
$\mathcal{L}_{V}$ becomes

$\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda^{2}_{W})}{8(1-3\lambda)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}\right.$
(11)
$\displaystyle\left.-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)+\frac{\Omega\kappa^{2}\mu^{2}}{8(3\lambda-1)}R\right\\}.$

The last term has been introduced in Horava:2009uw ; Nastase2009 ;
Kehagias:2009is . In the action, $\lambda,\kappa,\mu,\omega$,$\Lambda_{W}$ and
$\Omega$ are the coupling parameters, and $C_{ij}$ is the Cotton tensor
defined by

$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{j}_{l}-\frac{1}{4}R\delta^{j}_{l}\right).$
(12)

The study of the perturbations around the Minkowski vacuum shows that there is
ghost excitation when $\frac{1}{3}<\lambda<1$. This indicates that the theory
is only well-defined in the region $\lambda\leq\frac{1}{3}$ and $\lambda\geq
1$. Since the theory should be RG flow to IR with $\lambda=1$, we expect that
at UV, $\lambda>1$ to have a well-defined RG flow. At IR, $\lambda=1$, the
kinetic term recovers the one of standard general relativity. Comparing (II)
to the action of the general relativity in the ADM formalism, the speed of
light, the Newton’s constant and the cosmological constant emerge as

$\displaystyle
c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda_{W}}{1-3\lambda}},\hskip
12.91663ptG=\frac{\kappa^{2}}{32\pi c},\hskip
12.91663pt\Lambda=\frac{3}{2}\Lambda_{W}.$ (13)

It follows from (13) that for $\lambda>1/3$ ,the cosmological constant
$\Lambda_{W}$ has to be negative. It was noticed in Lu2009  that if we make an
analytic continuation of the parameters

$\mu\to i\mu,\hskip 17.22217pt\omega^{2}\to-i\omega^{2},$ (14)

the four-dimensional action (II) remains real as

$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})\right\\},$ $\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda^{2}_{W})}{8(3\lambda-1)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(3\lambda-1)}R^{2}\right.$
(15)
$\displaystyle\left.+\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\}.$

In this case, the emergent speed of light becomes

$c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda_{W}}{3\lambda-1}}.$ (16)

The requirement that this speed be real implies that $\Lambda_{W}$ must be
positive for $\lambda>\frac{1}{3}$.

##  III Electromagnetic Field in Hořava-Lifshitz Static Curved Spacetime

The electrodynamics in curved spacetime has been discussed many times in the
old days. Ellis(1973) first wrote down the Maxwell’s equations in $3+1$
congruence language. Later the Maxwell’s equations in $3+1$ form were fully
discussed by Thorne, Price, and Macdonald(1986). The Maxwell’s equations in a
curved spacetime could be written as

$\displaystyle\nabla_{\nu}F^{\mu\nu}=0,$ (18)
$\displaystyle\partial_{\mu}F_{\nu\rho}+\partial_{\nu}F_{\rho\mu}+\partial_{\rho}F_{\mu\nu}=0,$

where $F_{\mu\nu}$ is the antisymmetric electromagnetic tensor and
$F^{\mu\nu}=F_{\alpha\beta}g^{\alpha\mu}g^{\beta\nu}$. The electric field
$E_{i}$ and magnetic field $H_{i}$ are related to $F_{\mu\nu}$ as

$E_{i}=F_{ti},\;H_{i}=-\frac{\epsilon_{ijk}}{2}\sqrt{-g}F^{jk},$ (19)

where $E_{i}$ and $H_{i}$ are spatial vectors. The electromagnetic action in
the curved spacetime could be written in $3+1$ form

$S_{em}=\int
dtd^{3}\mathbf{x}\sqrt{h}Ng_{em}\left[F_{tk}F^{tk}+F_{kt}F^{kt}+F_{ij}F^{ij}\right],$
(20)

where $g_{em}$ is a constant. But the action at the Lifshitz point may be very
different. This has been discussed by HořavaHorava_Gugue_field , Chen and
Huang Chen_Huang_Lishitz_Point . In the paper Chen_Huang_Lishitz_Point , Chen
and Huang showed the action of Yang-Mills gauge field at the Lifshitz point in
the flat spacetime as

$S_{YM}=\frac{1}{2}\int
dtd^{d}\mathbf{x}\left[\frac{1}{g^{2}_{E}}Tr(E_{i}E_{i})-\sum_{J\geqslant
2}\mathcal{O}_{J}\star F^{J}\right],$ (21)

where

$\mathcal{O}_{J}=\frac{1}{g^{J}_{E}}\sum^{n_{J}}_{n=0}(-1)^{n}\frac{\lambda_{J,n}}{M^{2n+\frac{d+1}{2}J-d-1}}D^{2n}.$
(22)

Here $F$ and $D$ are the abbreviated denotation for $F_{ij}$ and $D_{k}$
respectively, and $\lambda_{J,n}$ are the the coupling with zero energy
dimension. Similarly $D^{2n}\star F^{J}$ also contains all possible
independent combinations of $D_{k}$ and $F_{ij}$. The action of Yang-Mills
theory at Lifshitz point in curved spacetime should be similar with action
show above. To the static charged black hole, $F_{tr}$ component of
$F_{\mu\nu}$ is not zero and it must be functions of $r$. If there are
independent magnetic charges in the world and the static black hole absorbs
some magnetic charges, the magnetic field component $H_{r}$ should not be
zero. From (19), $F^{\theta\phi}$ should not be zero. So the action of
electromagnetic field to static charged black hole will be reduced to

$S_{em}=\int dtd^{3}\mathbf{x}\mathcal{L}_{em}=\int
dtd^{3}\mathbf{x}\sqrt{h}N2g_{em}\left[F_{tr}F^{tr}+F_{\theta\phi}F^{\theta\phi}\right].$
(23)

From the equations (18) and (18) we get four independent equations

$\partial_{r}\left(\sqrt{h}NF^{tr}\right)=0.$ (24)
$\partial_{\theta}\left(\sqrt{h}NF^{\theta\phi}\right)=0,\;\;\;\partial_{\phi}\left(\sqrt{h}NF^{\theta\phi}\right)=0.$
(25)
$\partial_{r}F_{\theta\phi}+\partial_{\phi}F_{r\theta}+\partial_{\theta}F_{\phi
r}=0.$ (26)

##  IV Static Charged Black Hole Solutions

As the discussion in the first paragraph, we now seek the static charged black
hole solutions with the metric ansatz

$ds^{2}=-N(t,r)^{2}dt^{2}+\frac{1}{f(r)}(dr+N^{r}dt)(dr+N^{r}dt)+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (27)

With this metric ansatz, from (24), $F^{tr}$ should satisfy the equation

$\partial_{r}\left(\frac{N}{\sqrt{f}}r^{2}F^{tr}\right)=0.$ (28)

The solution of this equation is

$F^{tr}=\frac{Q_{e}\sqrt{f}}{Nr^{2}},$ (29)

where $Q_{e}$ is an integration constant. From (25) and (26), $F^{\theta\phi}$
and $F_{\theta\phi}$ should satisfy

$\partial_{\theta}\left(\frac{N}{\sqrt{f}}r^{2}\sin{\theta}F^{\theta\phi}\right)=0,\;\partial_{\phi}\left(\frac{N}{\sqrt{f}}r^{2}\sin{\theta}F^{\theta\phi}\right)=0,\;\partial_{r}F_{\theta\phi}=0.$
(30)

The solution of the three equations are

$F^{\theta\phi}=\frac{Q_{m}}{r^{4}\sin{\theta}},\;F_{\theta\phi}=Q_{m}\sin{\theta},$
(31)

where $Q_{m}$ is an integration constant. So from the action (23), the
Lagrangian of electromagnetic field is

$\mathcal{L}_{em}=-2g_{em}\frac{N}{\sqrt{f}}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(32)

The whole action of electromagnetic field couples with Hořava-Lifshitz gravity
is $S=S_{HL}+S_{em}$. Substituting the metric ansatz (27) into the Lagrangians
(II) and (23), up to an overall scaling constant, we get

$\displaystyle\mathcal{L}_{K}=$
$\displaystyle\frac{1}{N\sqrt{f}}\left\\{(1-\lambda)r^{2}f^{2}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)^{2}+2(1-2\lambda)f^{2}N_{r}^{2}\right.$
(33) $\displaystyle\hskip 8.61108pt\left.-4\lambda
rf^{2}N_{r}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)\right\\},$
$\displaystyle\mathcal{L}_{V}=$
$\displaystyle\frac{N}{\sqrt{f}}\left\\{2-3\Lambda_{W}r^{2}-2f-2rf^{{}^{\prime}}+\frac{\lambda-1}{2\Lambda_{W}}f^{{}^{\prime}2}\right.$
(34) $\displaystyle\hskip
8.61108pt\left.-\frac{2\lambda(f-1)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{W}r^{2}}\right\\},$
$\displaystyle\mathcal{L}_{em}=$
$\displaystyle-2\tilde{g}_{em}\frac{N}{\sqrt{f}}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(35)

where $N_{r}=N^{r}/f$ and ′ means derivative to $r$. Here we have set $c=1$
and $\tilde{g}_{em}=\kappa^{2}g_{em}/2$. The full Lagrangian is
$\mathcal{L}=\mathcal{L}_{K}+\mathcal{L}_{V}+\mathcal{L}_{em}$. By varying the
action with respect to the functions $N_{r}$ , $f$ and $N(t)$, we obtain three
equations of motions,

$\displaystyle 0=2(1-\lambda)r^{2}f^{2}\frac{1}{N}$
$\displaystyle\left\\{N_{r}^{{}^{\prime\prime}}+\frac{f^{{}^{\prime\prime}}}{2f}N_{r}+\frac{3}{2}\frac{f^{{}^{\prime}}}{f}N_{r}^{{}^{\prime}}+2\frac{N_{r}^{{}^{\prime}}}{r}+\frac{1-2\lambda}{1-\lambda}\frac{f^{{}^{\prime}}}{f}\frac{N_{r}}{r}-2\frac{N_{r}}{r^{2}}\right.$
(36)
$\displaystyle\left.-\frac{N^{{}^{\prime}}}{N}\left(N_{r}^{{}^{\prime}}+N_{r}\frac{f^{{}^{\prime}}}{2f}-\frac{2\lambda}{1-\lambda}\frac{N_{r}}{r}\right)\right\\},$
$\displaystyle 0$
$\displaystyle=-\left(\frac{f^{{}^{\prime}}}{2f}\frac{1}{N}+\frac{N^{{}^{\prime}}}{N^{2}}\right)\left\\{(1-\lambda)r^{2}fN_{r}\left(N_{r}^{{}^{\prime}}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)-2\lambda
rfN_{r}^{2}\right\\}$ (37)
$\displaystyle+\left(N^{{}^{\prime}}-\frac{f^{{}^{\prime}}}{2f}N\right)\left\\{-2r+\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime}}-\frac{2\lambda(f-1)}{\Lambda_{W}r}\right\\}+N\left\\{\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime\prime}}+\frac{2(1-\lambda)(f-1)}{\Lambda_{W}r^{2}}\right\\}$
$\displaystyle+\frac{1}{N}\left\\{(1-\lambda)r^{2}fN_{r}N_{r}^{{}^{\prime\prime}}+\frac{1}{2}(1-\lambda)r^{2}f^{{}^{\prime\prime}}N_{r}^{2}-(1-\lambda)r^{2}fN_{r}^{{}^{\prime}2}+(1-\lambda)r^{2}f^{{}^{\prime}}N_{r}N_{r}^{{}^{\prime}}\right.$
$\displaystyle+\left.2(1+\lambda)rfN_{r}N_{r}^{{}^{\prime}}+(1-\lambda)rf^{{}^{\prime}}N_{r}^{2}+(6\lambda-4)fN_{r}^{2}\right\\}+\frac{1}{2\sqrt{f}}\left\\{\mathcal{L}_{K}+\mathcal{L}_{V}\right\\}-\tilde{g}_{em}\frac{N}{f}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}},$
$0=-\mathcal{L}_{K}+\mathcal{L}_{V}+\mathcal{L}_{em}.$ (38)

###  IV.1 Solutions with Projectability Condition

For the projectability condition, lapse function $N$ only is a function of
$t$, $N^{{}^{\prime}}$ is zero and we can set $N(t)=1$ with the time
rescaling. Obviously for all $\lambda$, $N_{r}=0$ is a solution of (36). In
this case, we assume ansatz $f(r)=1+yr^{2}$, just when $Q=0$, from (37),(38),
we get quadratic equations of $y$,

$\displaystyle y^{2}-2\Lambda_{W}y-3\Lambda_{W}^{2}=0,$ (39) $\displaystyle
y^{2}+2\Lambda_{W}y+\Lambda_{W}^{2}=0.$ (40)

Their solution is $y=-\Lambda_{W}$. This solutions corresponds to a maximally
symmetric space with curvature $\Lambda_{W}$. This result has been shown in
paper static_black_hole_of_HL .

In the IR region where $\lambda=1$, the equation (36) is reduced to

$\frac{f^{{}^{\prime}}}{f}\frac{N_{r}}{r}=0.$ (41)

Its $N_{r}=0$ solution has been discussed above. It has another solution as
$f=constant$. when $f$ is a constant, we could get the same equation from
(37),(38) as

$(N_{r}^{2})^{\prime}+\frac{N_{r}^{2}}{r}+\frac{1}{2f^{2}}\left\\{-3\Lambda_{W}r+\frac{2(1-f)}{r}+\frac{(1-f)^{2}}{\Lambda_{W}r^{3}}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{3}}\right\\}=0.$
(42)

The solutions of this equation are

$N_{r}=\pm\frac{1}{f}\sqrt{\frac{M}{r}+\frac{\Lambda_{W}}{2}r^{2}+(f-1)+\frac{(1-f)^{2}}{2\Lambda_{W}r^{2}}-\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}}.$
(43)

Especially when $f=1$, the solution reduces to

$N_{r}=\pm\sqrt{\frac{M}{r}+\frac{\Lambda_{W}}{2}r^{2}-\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}}.$
(44)

It is a (A)dS-Reissner-Nordstrom black hole written in Painlevé-Gullstrand
type coordinates. Especially when $Q_{e}=Q_{m}=0$, it is (A)dS-Schwarzschild
black hole which has been shown in static_black_hole_of_HL . The electric
field of the black hole (44) is $E_{r}=F_{tr}=(-Q_{e})/r^{2}$. Its charge is

$Q_{BH}=\frac{1}{4\pi}\int_{S}\vec{E}\cdot
d\vec{\sigma}=\frac{1}{4\pi}E_{r}\cdot 4\pi r^{2}=-Q_{e},$ (45)

where $S$ is a closed surface everywhere with the same $r$ and $d\vec{\sigma}$
is surface integral element. We should choose $\tilde{g}_{em}=1$ when the
solution (44) is a charged black hole with charge $\pm Q_{e}$.

In the UV region where $\lambda\neq 1$, when $f$ is a constant, from (36),(37)
and (38), we get three equations,

$\displaystyle 0=$ $\displaystyle
N_{r}^{{}^{\prime\prime}}+2\frac{N_{r}^{{}^{\prime}}}{r}-2\frac{N_{r}}{r^{2}},$
(46) $\displaystyle 0=$
$\displaystyle(1-\lambda)r^{2}N_{r}^{{}^{\prime}2}-4\lambda
rN_{r}N_{r}^{{}^{\prime}}+2(1-2\lambda)N_{r}^{2}-\frac{4(1-\lambda)(f-1)}{\Lambda_{W}r^{2}f}$
(47)
$\displaystyle-\frac{1}{f^{2}}\left\\{2(1-f)-3\Lambda_{W}r^{2}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{2}r^{2}}\right\\}+2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{f^{2}r^{2}},$
$\displaystyle 0=$ $\displaystyle(1-\lambda)r^{2}N_{r}^{{}^{\prime}2}-4\lambda
rN_{r}N_{r}^{{}^{\prime}}+2(1-2\lambda)N_{r}^{2}$ (48)
$\displaystyle-\frac{1}{f^{2}}\left\\{2(1-f)-3\Lambda_{W}r^{2}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{2}r^{2}}\right\\}+2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{f^{2}r^{2}}.$

Just when $f=1$ and $Q_{e}^{2}-Q_{m}^{2}=0$, they have a solution as

$N_{r}=\pm\;\sqrt{\frac{\Lambda_{W}}{(3\lambda-1)}}r.$ (49)

This solution actually describes the same de-Sitter space-time. One easy way
to see this point is to change inversely into the Schwarzschild coordinates.
This result is the same as paper static_black_hole_of_HL .

###  IV.2 Solutions without Projectability Condition

Without projectability condition, $N$ is function of $r$, $N^{\prime}$ isn’t
zero. In the IR region where $\lambda=1$, the equation (36) is reduced to

$\left(2\frac{N^{{}^{\prime}}}{N}-\frac{f^{{}^{\prime}}}{f}\right)\frac{N_{r}}{r}=0.$
(50)

Its solutions are $N_{r}=0$ or $N^{2}=f$. In the case $N_{r}=0$, with the same
discussion above, we find a solution

$f=1-\Lambda_{W}r^{2},$ (51)

and the function $N(r)$ is unconstrained. This result is the same as paper
Lu2009 .

When $\lambda=1$, in the case $N^{2}=f$, from (37),(38), we get one same
equation,

$\displaystyle
0=2\left[rfN_{r}^{2}\right]^{{}^{\prime}}+\left\\{2(1-f)-3\Lambda_{W}r^{2}-2rf^{{}^{\prime}}+\frac{2(1-f)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(f-1)^{2}}{\Lambda_{W}r^{2}}\right\\}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(52)

This equation has a solution as

$N_{r}^{2}=\frac{\beta}{rf},\;f=1-\Lambda_{W}r^{2}-\sqrt{cr+2\tilde{g}_{em}\Lambda_{W}\left(Q_{e}^{2}-Q_{m}^{2}\right)},$
(53)

Where $\beta,c$ are integration constants. If we reconsider the action (11)
which contains the “soft” violation term, we could get a equation similar as
(52) as

$\displaystyle
0=2\left[rfN_{r}^{2}\right]^{{}^{\prime}}+\left\\{-2\frac{\Omega-\Lambda_{W}}{\Lambda_{W}}(1-f-rf^{{}^{\prime}})-3\Lambda_{W}r^{2}+\frac{2(1-f)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(f-1)^{2}}{\Lambda_{W}r^{2}}\right\\}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(54)

Its solution is

$N_{r}^{2}=\frac{\beta}{rf},\;f=1+\left(\Omega-\Lambda_{W}\right)r^{2}-\sqrt{\Omega(\Omega-2\Lambda_{W})r^{4}+cr+2\tilde{g}_{em}\Lambda_{W}\left(Q_{e}^{2}-Q_{m}^{2}\right)},$
(55)

Where $\beta,c$ are integration constants. When $Q_{m}=0$, this solution is
similar as paper RongCai-2009  and When $Q_{e}^{2}-Q_{m}^{2}=0$ it is the
similar as park2009 . The solutions (53),(55) are two especial static charged
black hole solutions.

In the UV region where $\lambda\neq 1$, when $N_{r}=0$, from (37).(38), we get
two equations,

$\displaystyle 0=$ $\displaystyle
2-3\Lambda_{W}r^{2}-2f-2rf^{{}^{\prime}}+\frac{\lambda-1}{2\Lambda_{W}}f^{{}^{\prime}2}-\frac{2\lambda(f-1)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{W}r^{2}}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}},$
(56) $\displaystyle 0=$
$\displaystyle\left(N^{{}^{\prime}}-\frac{f^{{}^{\prime}}}{2f}N\right)\left\\{-2r+\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime}}-\frac{2\lambda(f-1)}{\Lambda_{W}r}\right\\}+N\left\\{\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime\prime}}+\frac{2(1-\lambda)(f-1)}{\Lambda_{W}r^{2}}\right\\}$
(57)

They have two new solutions as

$Q_{e}^{2}-Q_{m}^{2}=0,\;f=1-\Lambda_{W}r^{2}-\alpha
r^{\frac{2\lambda\pm\sqrt{6\lambda-2}}{\lambda-1}},\;N=ar^{-\frac{1+3\lambda\pm
2\sqrt{6\lambda-2}}{\lambda-1}}\sqrt{f},$ (58)

where $\alpha,a$ are constants. The solutions has been got by paper Lu2009 .

## Acknowledgments

The work was partially supported by NSFC Grant No.10535060, 10775002, 10975005
and RFDP. I would like to thank Bin Chen for drawing my attention to Hořava-
Lifshitz gravity and giving me many useful suggestions.

## References

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# Stretched Exponential Relaxation

Dan Elton  Daniel C. Elton

(11/2013
published on arXiv )

##  1 The stretched exponential

Figure 1: The stretched exponential on a log plot for various values of
$\beta$

The stretched exponential is:

$\phi(t)=A\exp\left[-\left(\frac{t}{\tau_{\mbox{\footnotesize{str}}}}\right)^{\beta}\right]$
(1)

Rudolf (Hermann Arndt) Kohlrausch ( 1809 - 1858) first proposed the stretched
exponential in 1854 to describe the relaxation of charge from a glass Leiden
jar.[11]111Historical aside : unfortunately a large number of works (about 2/3
of citations) do not provide the correct citation to R. Kohlrausch’s work.[2]
The error seems to have stemmed from an article by Palmer, Stein, & Abrahams
in 1984 where they improperly cited an 1847 article by R. Kohlrausch.[2] In
his 1847 article R. Kohlrausch describes qualitatively the torsional
relaxation of galvanometer threads but does not mention the stretched
exponential expression.[10] The confusion on this may have stemmed from the
fact that R. Kohlrausch’s son, Friedrich (Wilhelm Georg) Kohlrausch
(1840-1910) used the stretched exponential to describe the same mechanical
relaxations, but not until a later paper in 1863.[9] (Friedrich Kohlrausch
also used the stretched exponential to describe mechanical stretching/creep
relaxations in the natural polymer silk in 1863 and 1866.) Thus the first
mention of the stretched exponential in the literature is in 1854, not
1847.[11] The second documented use of the stretched exponential appears to be
by Werner in 1907 to describe the luminescence decay of an organic
phosphor.[17] In his article he does not cite Kohlrausch, suggesting his use
of the stretched exponential was independent of Kohlrausch’s. However, as was
recently pointed out, whereas Kohlrausch’s work has received thousands of
citations, Werner’s work has received only 8 SCI citations.[1]. The Williams-
Watts work in 1970 does not provide any citations and is done seemingly
without any knowledge of earlier uses, thus lending some justification to the
immense number of citations (2000+) their paper gained. The term “stretched
exponential” does not appear in the literature until 1984.

He came to expression 1 by assuming that the decay rate $k=1/tau$ was not
constant but deceased with time as $t^{\beta-1}$.

The stretched exponential has an exceptionally wide range of application
throughout science. In the materials science and engineering literature, an
expression known as the “Weibull distribution” is used , with $\beta$ taking
any value. (For $\beta=2$ the Gaussian distribution is obtained) In the
social/economic sciences it is used to describe things such as urban
population sizes, currency exchange rate fluctuations, and the success of
Hollywood blockbusters, musicians and scientists. (Since it has a fat tail,
like the power law it is useful in describing “rich get richer” phenomena
which are widespread in the social sciences). In 1993 Bömer listed stretching
exponents for over 70 materials, obtained by viscoelastic, calorimetric,
dielectric, optical, and other response measurements.

In mathematics, the stretched exponential is the characteristic function
(Fourier transform of) a type of stable distribution called the (Lévy)
“symmetric alpha-stable distribution”. (There is no general analytic
expression for the Fourier transform of the stretched exponential, so this
class of distributions is defined simply as the Fourier transform of the
stretched exponential - this point is discussed further below).

In the field of dielectric spectroscopy it is often refereed to as the
Williams-Watts (WW) or Kohlrausch-Williams-Watts (KWW) function after a paper
by Williams & Watts in 1970.[18]

In all of its many applications, the stretched exponential distribution never
gives the entire picture, but as a phenomenological function it can often
describe data over many orders of magnitude. The fact that it contains only
one extra parameter, $\beta$, makes it exceptionally easy to fit to data, but
as we will see it does not have clean mathematical properties like the
Gaussian or exponential function, both of which are special cases of the
stretched exponential. The fact that such a simple analytic expression can fit
data over many orders of magnitudes suggests, but by no means proves, that it
must be of some fundamental significance.222However , there is probably not a
single model explaining the presence of the stretched exponential in all these
cases. This is an example of a common fallacy which physicists fall into -
which is to assume that things described by the same mathematical structure
have the same physical origin. As one example, Newtonian gravitation and
electrostatics are often said to have the same “mathematical structure” and
thus students are left with the impression that they are somehow the same “in
some deep way” when in fact they are vastly different both in theory,
practice, and origin (apart from the obvious difference in the strength of the
two forces, their origins are vastly different - one comes from taking the
limit of large energy density (lots of photons), the other comes from taking
the limit of small energy density (low gravitational fields)). As a more
extreme example, power laws can be used to describe the distributions of city
sizes, word frequencies and earthquake magnitudes (to name just a few things),
yet it would be silly to assume there is common “physics” behind them (other
than the fact that a power law describes them). In this case many disparate
systems are described by the stretched exponential, so it does not seem likely
that a common physical mechanism (or a single physical model) can be used to
explain the presence of this behavior for all of them (in the opinion of this
author). Still, just as the power law is generally indicative of the presence
of self-similarity, there are likely common physical features associated with
the stretched exponential in a very universal way. In the field of glasses
though, it can be said with good confidence that the stretched exponential has
some truly fundamental significance. Even the simplest theoretical model of a
glass - the 3D $\pm J$ spin glass on a cubic lattice - exhibits stretched
exponential relaxation in the spin-spin autocorrelation function, as has now
been conclusively shown by computer simulation.333This could be a point of
contention because the spin glass model does not contain dynamics – they must
be added on top. Most of the rigorous work on spin glasses is focused on
exploring their energy landscape and equilibrium properties rather than their
dynamics. There are many ways of adding dynamics – for one such dynamics – the
“Glauber dynamics” – theoretical arguments predict a power law decay.[14]. The
result quoted by Johnston is from “molecular dynamics simulation” which give a
“overall time behavior” of $s(t)\propto t^{-x}\exp[-(t/\tau)^{\beta}]$, with
the prefactor $t^{-x}$ being inconsequential for $t\gg 0$.[15] The classic
reference is Ogielski, 1985 – computer simulations published before that were
plagued by finite size effects due to lattice waves and did not show stretched
exponential relaxation.[15] At least $16^{3}=4096$ spins are required to get
stretched exponential relaxation.[15] A nearly linear decrease in $\beta(T)$
as a function of temperature is observed - decreasing from $\beta=1$ at the
Curie point $\approx 4.5T_{g}$ to $\beta=.35$ at the glass transition
temperature $T_{g}$.

In the studies of relaxation of glasses, theoretical justification can be
found in the so-called “trapping model”, which was first developed in the
1970s by mathematical physicists to describe the non-radiative recombination
of excitons in amorphous semiconductors. The mathematical intricacies of the
trapping model are quite complex, but the basic idea is rather simple to
visualize. It is assumed that the material is filled with randomly distributed
traps. Traps capture excitations whenever they are within a certain radius of
the trap. During relaxation, excitons (or more generally, molecules) diffuse
through the material and fill these traps. As the traps are filled, the
remaining excitons must travel for longer to find unfilled traps, thus
explaining the $k=t^{\beta-1}$ decrease in the rate. In particular, the
trapping model predicts:

$\beta=\frac{d}{d+2}$ (2)

In three dimensions, the trapping model gives $\beta=3/5=.6$. There have been
many other attempts to give the stretched exponential a theoretical
justification, mostly in the field of glasses and disordered materials – a
review of these theories and the related experimental data was given by J.C.
Phillips in 1996, with the focus mainly on the trapping model.[15] According
to Phillips the trapping model is the only model which describes the emergence
of the stretched exponential in truly microscopic terms and does so with
mathematical rigor (albeit only with great effort, requiring concepts from
axiomatic set theory).[15] Furthermore, Phillips argues that the trapping
model is much more widely applicable than was previously thought.

Klafter identifies three separate models which yield a stretched exponential
relaxation – the Förster energy transfer model, the hierarchically constrained
dynamics picture and the “defect diffusion model” – and he shows that all
three share mathematical similarities.[8]

Models yielding stretched exponential relaxation can be divided into two types
- serial and parallel.[3] (Additionally, there are references in the
literature suggesting that it might necessary to combine both). In a serial
model, relaxation proceeds through a number of steps separated by barriers of
different heights. In a parallel model, each molecule is relaxing
exponentially through a single barrier, but there are different barrier
heights for different molecules.

The author of this article has studied dielectic relaxation in water with
molecular dynamics simulation, noting how the rotational relaxation of
molecules follows a stretched exponential form.[4] Possible serial and
parallel explanations for this were explored.[4] Counter intuitvely, the
overall dielectric relaxation of water is largely exponential,[5] but
interestly the dielectric relaxation becomes stretched exponential in nano-
confined water.[16]

###  1.1 Mathematical properties of the stretched exponential

####  1.1.1 $\tau$ distribution

Figure 2: $G(k)$ distribution for various values of $\beta$. Figure 3: Same
data as in fig. 2 but showing $G(\tau)$ for various values of $\beta$. (It
appears that the normalization was lost during the transformation)

The stretched exponential can be understood as a continuous sum of exponential
decays via the following transformation:

$\exp\left[-\left(\frac{t}{\tau_{\mbox{\footnotesize{str}}}}\right)^{\beta}\right]=\int_{0}^{\infty}P(s,\beta)\exp(-s\frac{t}{\tau_{\mbox{\tiny{str}}}})ds$
(3)

Here $s$ is defined as the dimensionless variable $s\equiv
k/k_{\mbox{\footnotesize{str}}}=\tau_{\mbox{\footnotesize{str}}}/\tau$.

Equation 3 is an example of a Laplace transform, thus, to recover $P(s,\beta)$
requires taking the inverse Laplace transform:

$P(s,\beta)=\frac{1}{2\pi i}\int_{\Gamma}e^{-x^{\beta}}e^{sx}dx$ (4)

Where $\Gamma$ is a path from $-i\infty$ to $i\infty$ such that all the
singularities in the integrand lie on the left.

A change of variables $u=ix$ allows one to write this as a Fourier transform:

$P(s,\beta)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-(iu)^{\beta}}e^{isu}du$
(5)

$P(s,\beta)$ is a probability distribution normalized to one. Mathematically,
these distributions are asymmetric Lévy stable distributions.[7] (Not to be
confused with the symmetric Lévy stable distributions, of which the stretched
exponential is the characteristic function). (More precisely stated, there is
a one-to-one correspondance between the asymmetric Lévy stable distributions
and the space of functions of the form $\exp((kt)^{\beta})$ where
$\beta\in(0,2)$.) Just as the centralized limit says that idd random variables
sampled from distributions with finite variance will eventually converge to a
Gaussian, the generalized centralized central limit theorem says that iid
variables taken from distributions with power law tails of the form
$|x|^{-\beta-1}$ ($\beta\in(0,2)$) will lead to Asymmetric Lévy stable
distributions (since power law tails are not normalizable, a special
normalization factor $1/N^{1/\beta}$ must be used to construct the limiting
distribution).[12] In light of this fact the ubiquitous nature of the
stretched exponential is not as surprising, since power law distributions are
also ubiquitous.

A few different formulae have been derived for calculating $P(s,\beta)$ for
arbitrary $\beta$. One is a summation formula:

$P(s,\beta)=\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\Gamma(n\beta+1)}{n!s^{n\beta+1}}\sin(n\pi\beta)$
(6)

The convergence of this series appears to be very slow – to produce good plots
$\approx$ 10,000 terms were required, thus making use of this formula very
cumbersome even on a fast machine. A better equation can be obtained by
looking at the real and imaginary parts of equation 5. It can be shown that
the imaginary part is exactly zero (which is good because imaginary
probabilities don’t make much sense). The real part is:[6]

$P(s,\beta)=\frac{1}{\pi}\int_{0}^{\infty}e^{-u^{\beta}\cos(\pi\beta/2)}\cos[su-u^{\beta}\sin(\pi\beta/2)]du$
(7)

A rigorous derivation of this formula is given in [13] using different
notation. Equation 7 evaluates much faster in Mathematica (production of a
nice looking plot of $P(s,\beta)$ at $\approx$ 1000 points takes about 1.5
minutes distributed over four 2.2 GHz i7 cores).

It is also possible to obtain so-called “closed form expressions” (in terms of
Airy, Bessel, Gamma, hypergeometric and complex error functions) from for
rational values of $\beta=n/m$, although these expressions rapidly increase in
complexity as with larger $m$. A particularly nice closed form expression is
obtained for $\beta=1/2$:

$P(s,\frac{1}{2})=\frac{1}{\sqrt{4\pi s^{3}}}\exp\left(-\frac{1}{4s}\right)$
(8)

The case $\beta=1/2$ was studied exclusively by Williams & Watts in their
seminal paper.

####  1.1.2 Behavior near zero

Near zero the stretched exponential exhibits a singularity in its derivative.
In fact, this singularity appears at all higher orders of derivatives as well.
When $t\ll\tau_{\mbox{\footnotesize{str}}}$ one finds

$s\phi(t)\approx
A-A\beta\left(\frac{t}{\tau_{\mbox{\footnotesize{str}}}}\right)^{\beta}$ (9)

###  1.2 Properties of $G(k)/G(\tau)$

The properties and limiting behavior of $P(s,\beta)$ are studied in detail by
Johnson in his work to better understand how to interpret both $\beta$ and
$k_{\mbox{\footnotesize{str}}}$.[6] We will not go through his detailed
analysis here, but will note some of the key results because they are very
illuminating.

  * •

The max in $G(\tau)$ increases with increasing $\beta$ in a way which can be
fit by an exponential function

  * •

$G(k)$ contains a low-$k$ (high $\tau$) exponential cutoff that decreases with
decreasing $\beta$.

  * •

$G(k)$ contains a power law tail at high k described by :

$\displaystyle P(k\rightarrow\infty)$ $\displaystyle=\frac{c}{s^{1+\beta}}$
(10) $\displaystyle c$
$\displaystyle=\frac{\Gamma(\beta+1)}{\pi}\sin(\pi\beta)$

  * •

The first moment (average) of $G(k)$ is infinite

  * •

All higher moments of $G(k)$ are infinite

  * •

The moments of $G(\tau)$ are finite and are given by:

$\langle\tau^{m}\rangle=\frac{\Gamma(m/\beta)}{\beta\Gamma(m)}$ (11)

  * •

$k_{\mbox{\footnotesize{str}}}$ in the stretched exponential function is not
the average rate, nor is it the inverse of the average $\tau$. It is
approximately the median $k$, but only for $.5\beta 1$.

  * •

The $\beta$ factor cannot in any way be used to describe adequately the
“width” of the distribution $G(k)$. However it is related monotonically to the
low $k$ (large $\tau$) cutoff. From the average of $G(\tau)$ another
interpretation is that $1/\beta$ is a measure of the average relaxation time
relative to $\tau_{\mbox{\footnotesize{str}}}$.

###  1.3 Barrier height distributions

For a system where the higher energy state is separated from the lower energy
state by a barrier of height $E_{a}$, the relaxation time (assuming Arrhenius
behavior) is related to $E_{a}$ via:

$\displaystyle k$ $\displaystyle=k_{0}e^{-\frac{E}{kT}}=\tau_{0}e^{-\epsilon}$
(12) $\displaystyle E$ $\displaystyle=-kT\ln(k/k_{0})=-kT\ln(\tau_{0}/\tau)$

This allow one to transform $P(s,\beta)$ into $P(E_{a},\beta)$, which is a
distribution of barrier heights. A way of calculating these barrier height
distributions and approximating them by analytic functions was discussed by
Edholm & Blomberg in 2000.[3] These distributions look like skewed Gaussians,
with skewness towards the smaller energies. In the limit of $\beta\rightarrow
0$ the distribution becomes Gaussian, but the approach is not a smooth
approach - even for values of $\beta$ close to zero the distribution remains
very skewed. Edholm & Blomberg note that the approximate width of the barrier
height distribution is well described by

$\sigma=1/\beta$ (13)

for $0<\beta<.85$. They also note that for $0<\beta<.85$ the distribution can
be somewhat approximated by a Gaussian and analyze how good this approximation
is. One one might therefore wonder what happens if we just assume a Gaussian
barrier height distribution. Unfortunately this leads to taking the inverse
Laplace transform of an expression of the form $\exp(\exp(x^{2}))$, for which
no closed form expression is known.

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