Table Of ContentAn algebraic construction of duality functions for the stochastic
(1)
(A ) vertex model and its degenerations
q n
U
Jeffrey Kuan
7
Abstract
1
0 Arecentpaper[KMMO16]introducedthestochasticU (A(1))vertexmodel. ThestochasticS–matrix
q n
2 isrelatedtotheR–matrixofthequantumgroupU (A(1))byagaugetransformation. Wewillshowthat
q n
n a certain function D+ intertwines with the transfer matrix and its space reversal. When interpreting
µ
a thetransfermatrixasthetransitionmatrixofadiscrete–timetotallyasymmetricparticlesystemonthe
J one–dimensional lattice Z, the function D+ becomes a Markov duality function D which only depends
µ µ
1 on q and the vertical spin parameters {µ }. By considering degenerations in the spectral parameter,
x
3 the duality results also hold on a finite lattice with closed boundary conditions, and for a continuous–
timedegeneration. Thisdualityfunctionhadpreviouslyappearedinamulti–speciesASEP(q,j)process
]
[Kuan16]. TheproofhereusesthattheR–matrixintertwineswiththeco–product,butdoesnotexplicitly
R
use the Yang–Baxter equation.
P
It will also be shown that the stochastic U (A(1)) is a multi–species version of a stochastic vertex
. q n
h modelstudiedin[BorPet16,CorPet16]. Thiswillbedonebygeneralizingthefusionprocessof[CorPet16]
t and showing that it matches the fusion of [KuReSk81] up to the gauge transformation.
a
m We also show, by direct computation, that the multi–species q–Hahn Boson process (which arises
at a special value of the spectral parameter) also satisfies duality with respect to D , generalizing the
0
[
single–species result of [Cor15].
2
v
Contents
8
6
4 1 Introduction 2
4
0 2 Preliminaries 6
. 2.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1
2.1.1 q–notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0
2.1.2 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7
1 2.1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
: 2.1.4 Lumpability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
v
2.1.5 Operator Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
i
X 2.2 Results from [KMMO16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The q–Hahn Boson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
r
a 2.4 Results from [Kuan16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Relationship to previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.1 The [CGRS15] framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.2 Single–species stochastic vertex model from [BorPet16, CorPet16]. . . . . . . . . . 16
3 Further results about S(z) 17
3.1 Additional Degenerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 At l=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.2 At m=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.3 At z=0,z→∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Stochastic fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1
3.3 Stochasticity of S(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Lumpability of S(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Analytic Continutation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.1 The single–species n=1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Intertwining of transfer matrices 30
4.1 Equivalent expression for duality function . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 The one–site case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Extension to L sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Interpretation as a particle system satisfying duality . . . . . . . . . . . . . . . . . . . . . 35
4.4.1 On the infinite line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.2 Closed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4.3 Continuous–time zero range process . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Descriptions of processes 40
5.1 At z=ql−m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 l=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.1 Discrete–time process with blocking . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.2 Continuous–time zero–range process . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 m=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 Conjecture for general l,m,z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 “Direct” results for multi–species q–Hahn Boson 43
6.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 The Duality Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.3 Lumpability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A Explicit examples 50
A.1 Fusion for l=1,m=2,n=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.2 S(z) for l=m=2,n=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1 Introduction
Overthelast25years,muchworkhasbeendoneinvestigatinginteractingparticlesystemswithaprop-
erty called stochastic duality (e.g. [BarCor16, BelSch15, BelSch15-2, BelSch16, BorCor13, BorCorSa14,
CGRS15,CGRS16,Cor15,GKRV09,GRV10,Kuan15,Kuan16,Ohku16,Sch97,Sch16,SchSan94]). Du-
ality has been shown to be useful for asymptotics [BorCorSa14], weak convergence [CST16], and shock
dynamics [BelSch16]. The first use of duality in interacting particle systems actually goes back even
farther, to 1970 [Spit70]. See also Chapter III of [Ligg] for an exposition and references.
A more recent direction of research has been to find dualities in multi–species versions of some of
thesesystems. Thiswasdonein[BelSch15,BelSch15-2,BelSch16,Kuan15,Kuan16]. Inthesecases,the
interactingparticlesystemsatisfiedaLiealgebrasymmetry,andtherankoftheLiealgebracorresponded
to the number of species of particles.
Duality has also been discovered in stochastic vertex models (see e.g. [BorCorGor16, BorPet16,
CorPet16]). Thesevertexmodelsenjoythepropertyofdegeneratingtoalargeclassofotherprobabilistic
models, including some of the ones above. However, the proofs of these dualities seem to be ad–hoc, in
the sense that they required knowing the duality function beforehand, and did not involve constructing
the duality function using the algebraic symmetry of the model. Furthermore, there were no known
examples of multi–species vertex models satisfying duality.
Thus, it is natural to look for a stochastic vertex model such that
(1) self–duality holds with respect to a duality function which can be defined with the algebraic
symmetry of the vertex model, and
(2) in certain degenerations of the vertex model, previous duality results can be recovered, and
(3) is a multi–species generalization of an existing single–species model.
ThepurposeofthispaperistoprovethatthestochasticU (A(1))vertexmodelof[KMMO16]satisfies
q n
all three of these properties.
2
ThestochasticU (A(1))vertexmodelisdefinedfromastochasticmatrixS(z)dependingonaspectral
q n
parameter z; see section 2.2 for definitions. The matrix S(z) is obtained from the (non–stochastic) R–
matrix of Uq(A(n1)) by a gauge transform. The action of S(z) on a certain representation Vlz1 ⊗Vmz2
then defines a local Markov operator. Here, l is the horizontal spin parameter, m is the vertical spin
parameter, and the spectral parameters z and z satisfy z =z /z . The corresponding transfer matrix
1 2 1 2
then defines a Markov operator for an interacting particle system on a one–dimensional lattice.
The original paper [KMMO16] specifically considers z = ql−m and finds a n–species version of the
q–Hahn Boson process introduced in [Pov13]. The q–Hahn Boson process has both a discrete–time and
a continuous–time definition, with the continuous–time process satisfying a Hecke algebra symmetry, as
shown in [Take14]. When the vertical spin parameter µ=q−m converges to 0, this further degenerates
toasinglespeciesq–Bosonprocessintroducedby[SaWa98]. Thepaperherewillconsidergeneralvalues
of z.
The main results will be summarized as follows.
(1)WewillshowthatthestochasticU (A(1))vertexmodel(forgenericvaluesofz)satisfiesself–duality
q n
with respect to a certain explicit duality function D . The function D had previously appeared as the
µ µ
duality function of a multi–species ASEP(q,j), and is defined from the action of a certain element u ∈
0
U (A ). It only depends on the vertical spin parameters µ and not on the horizontal spin parameter.
q n x
TheproofusesthattheR–matrixintertwineswiththeactionofU (A(1)),butdoesnotexplicitlyusethe
q n
Yang–Baxterequation. Theproofalsoinvolvesshowingthatthegaugeisthesameasthe“groundstate
transformation” of [Kuan16] up to a diagonal change of basis.
(2) For a range of values of the spectral parameter z, the matrix S(z) is stochastic (see Proposition
3.3). In section 3.1, some degenerations of S(z) will be considered. In particular, for degenerations
of z, the matrix S(z) becomes trivial (see Theorem 3.2). This actually allows for construction of the
particlesystemincontinuoustimeaswellasonafinitelatticewithclosedboundary(seesection4.4). A
noteworthy result is that the n–species q–Boson process introduced by [Take15] can be shown to satisfy
self–duality, which had previously been shown with different methods in [Kuan16]. Another interesting
case is when z=ql−m, which shows that the n–species discrete–time q–Hahn Boson process is self–dual
with respect to D . See Figures 1 and 2 for the degenerations discussed here.
µ
Italsoturnsoutthatthen–speciesdiscrete–timeq–HahnBosonprocessisalsoself–dualwithrespect
toD ,whichwasshownforn=1in[Cor15]. However,itisnotclearhowtoprovethisalgebraically. A
0
direct proof will be given.
(3) It will be shown that the stochastic U (A(1)) vertex model is a n–species generalization of the
q n
modelsof[CorPet16,BorPet16]. Thiswillbedonebyshowingthatthe“stochasticfusion”procedureof
[CorPet16]canbegeneralizedformultiplespecies(seeTheorem3.4). Additionally,theMarkovprojection
to the first k species is a U (A(1)) vertex model (see Proposition 3.8). The latter property is typical for
q k
multi–species models (see e.g. [Kuan16]).
It is also worth explicitly mentioning the role of the boundary conditions in the duality results. The
transfermatrixofthestochasticU (A(1))vertexmodelintertwineswithitsspacereversalunderacertain
q n
function D+ which also acts on the auxiliary space. In order to reduce D+ to a duality functional D
µ µ µ
whichdoesnotactontheauxiliaryspace,acertaincancellationisneeded(Lemma4.2),butthisreduction
does not seem to hold for open or periodic boundary conditions.
The remainder of the paper is outlined as such: Section 2 states the necessary definitions, notations
andresultsfrompreviouspapers. Insection3,furtherpropertiesofS(z)willbeproved,includingranges
for which the matrix is stochastic (Proposition 3.6), degenerations (section 3.1), Markov projections
(Theorem 3.8), and stochastic fusion (Theorem 3.4).
Section4definesthetransfermatrixandprovesdualityresultsfortheresultingparticlesystems. The
main theorem is stated in Theorem 4.5, which shows that D+ intertwines between the transfer matrix
µ
and its space reversal. This results in duality results for a discrete–time particle system on the infinite
line (Theorem 4.10), on a finite lattice with closed boundary conditions (Proposition 4.12), and for a
continuous–timedegeneration(Proposition4.14). Section5describestheprocessesthatcanbeobtained
from the various degenerations: subsection 5.1 considers the multi–species q–Hahn Boson process, and
section 5.2 considers the case when l=1. Section 6 shows, using direct computation, the self–duality of
multi–species q–Hahn Boson with respect to D , as well as the Markov projection property.
0
Acknowledgments. The author would like to thank Alexei Borodin and Ivan Corwin for helpful
conversations. FinancialsupportwasprovidedbytheMinervaFoundationandNSFgrantDMS–1502665.
3
(1)
Stochastic (A ) vertex model
q n
U
S(z) acts on V V
l m
⊗
Multi–species: [KMMO16]
Single–species: [CorPet16]*,[BorPet16]*
l=1,m=1
z =ql−m,λ:=q−l,µ:=q−m
Stochastic six vertex model
Discrete-time q-Hahn Boson Multi–species: —
Multi–species: — Single–species: [GwaSpo92],[BorCorGor16]
Single–species: [Pov13],[Cor15]*
λ 1
→ ASEP(q,j)
Multi–species:[Kuan16]*
Continuous–time q–Hahn Boson Single–species: [CGRS15]*
z q
Multi–species: — →
Single–species:[Take14]
j j =1/2
→∞
µ=0
q–Boson ASEP
Multi–species: [Take15] Multi–species: [Ligg76],[BelSch16]*,[BelSch15-2]*,[Kuan15]*
Single–species: [SaWa98] Single–species: [Spit70],[Sch97]*
Figure 1: The various degenerations and limits explicitly mentioned in this paper. A * indicates a paper
with a duality result. See Figure 1 of [CorPet16] for a more complete diagram. Additional degenerations
will be shown in Figure 2.
4
(1)
Stochastic (A ) vertex model
q n
U
S(z) acts on V V
l m
⊗
l=1
Figure 1
Discrete–time q–Boson
with particle constraint Discrete-time q-Hahn Boson
and blocking by class
m→∞ z→∞ (0<q<1) λ=q−1
z=0 (q>1) z =q1−m µ=q−m
Trivial Figure 1
Discrete–time q–Boson
with particle constraint No Blocking
and strong blocking
Continuous-time q-Hahn Boson
m
→∞
Discrete–time q–Boson
without particle constraint
and strong blocking Figure 1
Id
−
(Continuous–time) q–Boson
Multi–species: [Take15]
Single–species: [SaWa98]
Figure 2: Some new degenerations developed in this paper. The limit z is expressed as w 0, but
x
→ ∞ →
the set of x such that w =0 must be infinite. The degeneration labeled Id is described in section 4.4.3.
x
(cid:54) −
5
2 Preliminaries
This section will review results from previous papers and state necessary definitions.
2.1 Definitions
2.1.1 q–notation
For z∈C,q∈(0,1) and n∈N∪{∞}, let the q–Pochhammer symbol be defined by
(z;q) =(1−z)(1−zq)···(1−zqn−1).
n
Furthermore define
(cid:32) (cid:33)
n (q)
(q) =(q;q) , = n .
n n k (q) (q)
k n−k
q
Notice that
(q)
lim n =n!,
q→1(1−q)n
sothesecanbeviewedasq–deformationsoftheusualintegers,factorials,andbinomials. Anotheruseful
q–deformation is
qn−q−n (cid:20) n (cid:21) [n]!
[n] = , [n]! =[1] ···[n] , = q .
q q−q−1 q q q k [k]![n−k]!
q q q
Note that [n] =[n] , and similarly for the q–factorials and binomials.
q q−1
There is a q–analog of the Binomial theorem, which states that if A and B are variables such that
BA=qAB, then
l (cid:32) (cid:33)
(A+B)l =(cid:88) l AdBl−d.
d
d=0 q
Thiscanbestatedequivalentlyasasumoversubsets. AnysubsetL⊆{1,...,l}canbeidentifiedwitha
monomial in A and B by setting L index the locations of the variable A: for example, if L={2,3,5}⊆
{1,...,6}, then the corresponding monomial is BAABAB. For any subset L={r ,...,r }⊆{1,...,l}
1 d
with d elements, let
c (L)=|{(r,s):1≤r<s≤l and r∈L,s∈/ L}|.
d
Then
(cid:32) (cid:33)
(cid:88) qcd(L) = l . (1)
d
L⊆{1,...,l} q
|L|=d
For example, for l=4 and d=2, then
c ({1,2})=4, c ({1,3})=3, c ({1,4})=2, c ({2,4})=1, c ({3,4})=0, c ({2,3})=2
d d d d d d
and
(cid:32) (cid:33)
(1−q3)(1−q4) 4
1+q+2q2+q3+q4 = = .
(1−q)(1−q2) 2
q
Note that by the identity
(q;q)
(q) = ∞ ,
n (qn+1;q)
∞
it is possible to extend these q–deformations to complex numbers. Define the q–Gamma function Γ (z)
q
as
(q;q)
Γ (z)=(1−q)1−z ∞
q (qz;q)
∞
for |q|<1. When q →1, Γ (z) converges to the usual Gamma function Γ(z). This definition is related
q
to [n]! via
q
[n]! =qn(n−1)/2Γ (n+1).
q q−2
Theright–hand–sideiswell–definedevenifn∈/ N,sotheq–factorialsandbinomialsarestillwell–defined.
6
2.1.2 Representation Theory
The Drinfeld–Jimbo quantum affine algebra (without derivation) Uq(A(n1)) = Uq(s(cid:98)ln+1) is generated by
e ,f ,k ,0≤i≤n with the Weyl relations
i i i
k−1k =k k−1 =1, [k ,k ]=0,
i i i i i j
kiej =q2δij−δi,j−1−δi,j+1ejki, kifj =q−2δij+δi,j−1+δi,j+1fjki,
k −k−1
[e ,f ]=δ i i ,
i j ij q−q−1
and the Serre relations (for n≥2)
e2e −(q+q−1)e e e +e e2 =0
i i+1 i+1 i i+1 i+1 i
f2f −(q+q−1)f f f +f f2 =0
i i+1 i+1 i i+1 i+1 i
[e ,e ]=[f ,f ]=0, j (cid:54)=i±1
i j i j
where the indices are taken cyclically (i.e. as elements of Z/(n+1)Z). The co–product is an algebra
homomorphism ∆:U (A(1))→U (A(1))⊗2 defined by
q n q n
∆k±1 =k±1⊗k±1, ∆e =1⊗e +e ⊗k , ∆f =f ⊗1+k−1⊗f .
i i i i i i i i i i i
The formula for the co–product is actually not canonical. Another choice of co–product is
∆ k±1 =k±1⊗k±1, ∆ e =e ⊗1+k−1⊗e , ∆ f =1⊗f +f ⊗k ,
0 i i i 0 i i i i 0 i i i i
which leads to essentially the same algebraic structure. The co–product ∆ satisfies the co–associativity
property, which says that as maps from U (A(1)) to U (A(1))⊗3,
q n q n
(id⊗∆)◦∆=(∆⊗id)◦∆.
Because of co-associativity, higher powers of ∆ can be defined inductively and unambiguously as alge-
brahomomorphisms ∆(L) :U (A(1))→U (A(1))⊗L by
q n q n
(cid:16) (cid:17)
∆(L) = id⊗∆(L−1) ◦∆
We will also use Sweedler’s notation:
∆(L)(u)=(cid:88)u ⊗···⊗u ,
(1) (L)
(u)
where each u is some element of U (A(1)).
(x) q n
There is an involution ω of U (A(1)) defined on generators by
q n
ω(k )=k−1 , ω(e )=f , ω(f )=e .
i n+1−i i n+1−i i n+1−i
It is straightforward to check from the Weyl and Serre relations that ω is indeed an automorphism, and
it is immediate from the definition that ω2 =id.
For l∈Z , let V be the vector space with basis indexed by the set
≥0 l
B(n+1) :={α=(α ,...,α )∈Zn+1 :α +...+α =l}.
l 1 n+1 ≥0 1 n+1
Thesuperscript(n+1)willbedroppedifitisclearfromthecontext. Fori∈{1,2,...,n+1}=Z/(n+1)Z,
defineˆi=(0,...,0,1,−1,...,0)∈Zn+1withthe1attheithpositionandthe−1atthe(i+1)thposition.
For z∈C, define the representation πz of U (A(1)) on V by
l q n l
πlz(ei)|α(cid:105)=zδi,0[αi]q|α−ˆi(cid:105), πlz(fi)|α(cid:105)=z−δi,0[αi+1]q|α+ˆi(cid:105), πlz(ki)=qαi+1−αi|α(cid:105). (2)
The parameter z is called the spectral parameter of the representation. Let Vz denote the vector space
l
of the representation πz. The subalgebra generated by e ,f ,k ,1 ≤ i ≤ n is denoted U (A ), and for
l i i i q n
7
this subalgebra the spectral parameter does not play a role. The vector space V is the l–th symmetric
l
tensor representation, which will be used in section 2.5 below.
For any l ≥ 0, let |Ω(cid:105) ∈ V be the vacuum vector. In other words, Ω is the basis vector indexed by
l
(0,...,0,l). Analogously, let capital alpha |A(cid:105) ∈ V denote the basis vector indexed by (l,0,...,0). In
l
general, |α(cid:105) will be interpreted as a particle configuration with an α number of ith species particles for
i
1≤i≤n. The(n+1)thspeciesofparticleswillbeconsideredholes. Fromtheviewpointofprobability
theory, it is somewhat unnatural to consider holes as being present in the state space. Because of this,
it will also be useful to define α¯ =(α ,...,α ) and |α|=α +...+α . Note that if α∈B(n+1), then
1 n 1 n m
α equals m−|α|. Thus, any expression E(α) depending on α∈B can be written as an expression
n+1 m
E(α¯,m) depending on α¯ and m. In particular, define the limit
lim E(α):= lim E(α¯,m),
m→∞ m→∞
where α¯ does not depend on m.
This definition can be extended to vector spaces and operators. For any m ∈ {0,1,...,}∪{∞},
define V¯ to be the vector space indexed by the set
m
B¯ ={(α ,...,α )∈Zn :α +...+α ≤m}.
m 1 n ≥0 1 n
If α∈B , then α¯ ∈B¯ , and thus any map M on V can also be defined as a map on V¯ . Extend the
m m m m
map M on V¯ to a map on V¯ by defining M to be zero outside of V¯ . With these definitions, given
m ∞ m
any sequence of maps M on V , define the limit lim M to be a map on V¯ , if the limit exists.
m m m→∞ m ∞
Given 1≤i≤j ≤n+1, also let α =α +...+α . If i>j, then set α =0.
[i,j] i j [i,j]
2.1.3 Duality
Recall the definition of stochastic duality:
Definition 2.1. Two Markov processes (either discrete or continuous time) X(t) and Y(t) on state
spaces X and Y are dual with respect to a function D on X×Y if
E [D(X(t),y)]=E [D(x,Y(t))] for all (x,y)∈X×Y and all t≥0.
x y
Ontheleft–hand–side,theprocessX(t)startsatX(0)=x,andontheright–hand–sidetheprocessY(t)
starts at Y(0)=y.
Anequivalentdefinition(forcontinuous–timeprocessesanddiscretestatespaces)isthatifthegener-
ator1 L ofX(t)isviewedasaX×Xmatrix,thegeneratorL ofY(t)isviewedasaY×Ymatrix,and
X Y
D is viewed as a X×Y matrix, then L∗ D = DL . For discrete–time chains with transition matrices
X Y
P and P also viewed as X×X and Y×Y matrices, an equivalent definition is
X Y
P∗D=DP .
X Y
If X(t) and Y(t) are the same process, in the sense that X=Y and L =L (for continuous time) or
X Y
P =P (for discrete–time), then we say that X(t) is self–dual with respect to the function D.
X Y
Suppose that X=Y=SI, where I ⊆Z is an interval and S is a countable set. If σ is an involution
of I such that σ(x+1)=σ(x)−1 for all x, then σ induces an involution σ∗ of SI by
(σ∗η)(x)=η(σ(x)) for η:I →S.
IfL =σ∗◦L ◦σ∗ andL∗ D=DL ,thenwesaythatX(t)satisfiesspace–reversedself–dualitywith
X Y X Y
respect to D.
Remark 2.2. Intheliterature,someauthorsdonotdrawadistinctionbetweenself–dualityandspace–
reversed self–duality. However, for the duality functions of interest here, a totally asymmetric process
cannotsatisfyself–duality,butitdoessatisfyspace–reversedself–duality(seetheremarksbeforePropo-
sition 2.6 of [Kuan16]). The terminology here is chosen to emphasize this distinction.
1Notethatinprobabilisticliterature,astochasticmatrixhasrowswhichsumto1,whereasinmathematicalphysicsliterature,
the columns sum to 1. This paper uses the latter definition. If the former definition were used, then the definition of duality
wouldbeLXD=DL∗Y.
8
Remark 2.3. Notethatifc(x,y)isafunctiononX×Ywhichis constantunderthedynamicsofX(t)
and Y(t), then c(x,y)D(x,y) is also a duality function. This will be used to simplify the expression for
D(x,y). Forthispaper,c(x,y)willbeafunctionwhichonlydependsonthenumberofparticlesofeach
species, which is a constant assuming particle number conservation. See [Ohku16] for an example of
duality functions on a lattice with open boundary conditions, in which this type of simplification is not
applicable.
2.1.4 Lumpability
LetT beaX ×X matrixandletP ={p(i)} beapartitionofX ,andP ={p(i)} apartitionofX .
0 0 1 0 0 i 0 1 1 i 1
Recall the convention that a matrix is stochastic if the columns (rather than the rows) sum to 1. Say
that T is lumpable (with respect to P and P ) if for all p(i) ∈P
0 1 2 1 1
(cid:88) T (x ,x )= (cid:88) T (x ,x(cid:48))
0 1 0 0 1 0
x1∈p(1i) x1∈p(1i)
whenever x ,x(cid:48) are in the same block p(j) ∈P . Define the P ×P matrix T¯ by setting T¯(p(j),p(i))
0 0 0 0 0 1 0 0 0 1
to be the quantity above.
Thecompositionoflumpablematricesisagainlumpable. Toseethis,IfT isaX ×X matrixwhich
1 1 2
is lumpable with respect to P and P , then for x ∈p(k),
1 2 0 0
(cid:88) (cid:88) (cid:88)
(T T )(x ,x )= T (x ,x )T (x ,x )
0 1 2 0 0 2 1 1 1 0
x2∈p(2i) x2∈p(2i)x1∈X1
(cid:88) (cid:88) (cid:88)
= T (x ,x )T (x ,x )
0 2 1 1 1 0
p1(j)∈P1x1∈p1(j)x2∈p(2i)
= (cid:88) (cid:88) T¯ (p(i),p(j))T (x ,x )
0 2 1 1 1 0
p1(j)∈P1x1∈p1(j)
= (cid:88) T¯ (p(i),p(j))T¯ (p(j),p(k))
0 2 1 1 1 0
p1(j)∈P1
=T¯ T¯ (p(i),p(k)).
0 1 2 0
This does not depend on the choice of x in p(k), so T T is lumpable with T T =T¯T¯.
0 0 0 1 0 1 0 1
This is a generalization of a lumped Markov process introduced in [KeSn76]. The condition that a
MarkovprocessislumpableissimplytheconditionthataprojectionofaMarkovprocessisstillMarkov.
Therearemoregeneralconditionsofinterest: forexample,[PitRog81]givesanintertwiningconditionin
whichtheprojectionisrandom. Inparticular,ifT isastochasticX ×X matrix,andΞisastochastic
0 0 0
X ×X matrix, Λ is a stochastic X ×X matrix, then define the X ×X matrix
0 1 1 0 1 1
T =ΛT Ξ.
1 0
If ΛΞ is the identity matrix on X , and T ,T satisfy the intertwining relation
1 0 1
ΛT =T Λ,
0 1
then Λ maps the Markov chain defined by T to a well–defined Markov chain defined by T .
0 1
It is not hard to see that the Pitman–Rogers relation is a generalization of lumpability. if X is a
1
partition P of X , pick an arbitrary element x(p)∈p for each p∈P . Then define Ξ and Λ by
0 0 0
Ξ(x,p)=1 , Λ(p,x)=1 .
x=x(p) x∈p
It is immediate that ΛΞ is the identity matrix on P . If T is lumpable with respect to P , then
0 0 0
(cid:88)
T (p ,p )=(ΛT Ξ)(p ,p )= T (x,x(p ))
1 1 2 0 1 2 0 2
x∈p1
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does not depend on the choice of x(p), and is the transition matrix of the lumped Markov chain. Fur-
thermore, the lumpability implies that for x∈q,
(cid:88) (cid:88)
ΛT (p,x)= T (y,x)= T (y,x(q))=ΛT ΞΛ(p,x)=T Λ(p,x).
0 0 0 0 1
y∈p y∈p
Remark 2.4. OneexampleofalumpableMarkovprocessisn–speciesASEP.Inthisprocess,thereare
n species of particles, and at most one particle may occupy a lattice site. One can think of each species
ashavingadifferentmass. Ifaparticleattemptstojumptoasiteoccupiedwithaheavierparticle,then
the jump is blocked. If a particle attempts to jump to a site occupied with a lighter particle, then the
two particles switch places. All left jumps occur with the same rate (independent of the species), and
likewise all the right jumps occur with the same rate (independent of the species). It is not hard to see
that the projection onto the first k species results in k–species ASEP, since each particle treats lighter
particles the same as holes. This model was first introduced in [Ligg76].
Amoregeneralmodelwouldallowthejumpratestodependonthespeciesoftheparticles. Inthiscase,
theprojectionontothefirstkspeciesisnolongerak–speciesASEP.See[Ka99,Kuan15]forexamplesof
multi–species ASEP which have jump rates depending on the species. With open boundary conditions,
severalmodels(see. e.g. [Can15,CFRV16,CGGW16,CoMaWi15,Man15,ManVie15,Uch08])havejump
ratesattheboundarieswhichdependonthespecies,withjumpratesinthebulkthatareindependentof
the species. See also [CGW16, PEM09] for multi–species ASEP on a ring, with jump rates independent
of the species.
2.1.5 Operator Notation
We introduce some notation for operators. Given two linear spaces V and W, a symbol of the form
M will denote an linear map with domain V ⊗W. In particular, let P be the permutation
VW VW
operator P :V ⊗W →W ⊗V defined by P (v⊗w)=w⊗v. Given an operator M from V ⊗W
VW VW
to itself, let M(cid:102)denote the reversed operator on W ⊗V:
M(cid:102)=PVW ◦M ◦PWV.
Given M on V ⊗W, let Mˇ :V ⊗W →W ⊗V be the map P ◦M.
VW
Suppose {V : m ≥ 0} is a family of vector spaces and for each m ≥ 0, M is an operator on V .
m m m
By abuse of notation, the subscript m in M will often be dropped. Given m ,...,m , the tensor
m 1 L
power V ⊗···⊗V will be denoted V(L). For 1 ≤ a ≤ b ≤ L, let V[a,b] denote V ⊗···⊗V .
m1 mL ma mb
Let M⊗L denote the operator M ⊗···⊗M on V(L). Given σ ∈S , let Pσ be the operator from
m1 mL L
V ⊗···⊗V to V ⊗···⊗V defined by
m1 mL mσ(1) mσ(L)
Pσ(v ⊗···⊗v )=v ⊗···⊗v
1 L σ(1) σ(L)
and note that
PσM⊗L =M⊗LPσ.
If G is an operator on V(L) and σ∈S is the reversal permutation σ(j)=L+1−j, let
L
G(cid:101)=Pσ◦G◦Pσ.
If R acts on V ⊗V for l,m≥0, then R is the action on the i,j component of the tensor product of
l m ij
V⊗L for 1≤i,j ≤L.
The –ket |α,β(cid:105) means |α(cid:105)⊗|β(cid:105), and similarly for the bra– (cid:104)α,β|. The Greek letters η and ξ will
denote multiple tensor products, e.g. |η(cid:105)=|η1,...,ηL(cid:105)=|η1(cid:105)⊗···⊗|ηL(cid:105).
As usual, M∗ denotes the transpose of M.
2.2 Results from [KMMO16]
For this section, α,γ ∈V and β,δ∈V , where 0≤l≤m.
l m
For every z∈C and l,m≥0, there is an R–matrix
R(z):Vz1 ⊗Vz2 →Vz1 ⊗Vz2,
l m l m
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