Volumen 41 (2007), p´aginas 117–142
On corepresentations of equipped posets and their differentiation
Claudio Rodr´ıguez Beltr´ an Alexander G. Zavadskij
Universidad Nacional de Colombia, Bogot´a
Abstract. Corepresentations of equipped posets over the pair (F, G) are intro duced and studied, whereF ⊂Gis a quadratic field extension. The reduction algorithmsdVII and Completion for corepresentations (being in some intuitive sense dual to the known algorithms VII and Completion for representations) are built and investigated, with some applications. The generalized short ver sions of Differentiations VII andVII for representations and corepresentationsd of equipped posets with additional relations are described.
Keywords and phrases: equipped poset, representation, corepresentation, matrix problem of mixed type, differentiation functor.
2000 Mathematics Subject Classification: 15A21, 16G20, 16G30, 16G60.
Resumen. Se introducen y estudian corepresentaciones de posets equipados sobre la pareja (F,G), donde F ⊂Ges una extensi´on cuadr´atica de campos.
Los algoritmos de reducci´onV IId y Completaci´on para corepresentaciones (que son en alg´un sentido intuitivo dual a los ya conocidos algoritmos V II y Com pletaci´on para representaciones) se construyen e investigan, con algunas apli caciones. Las versiones cortas generalizadas de las DiferenciacionesV IIyV IId para representaciones y corepresentaciones de posets equipados con relaciones adicionales se describen.
1. Introduction
In modern representation theory, the diagrammatic methods and matrix prob lems (i.e. classification problems of linear algebra) play an important role. The study of representations of finite dimensional algebras, quivers, vectroids, or ders, posets (inclusively those with additional relations or structures) leads in many cases to matrix problems (see for instance [2, 3, 7, 10]).
In particular, representations of equipped posets over the pair of fields of real and complex numbers (R,C) (introduced and studied in [9, 10, 11]) are reduced
117
to certain matrix problems of mixed type over this pair. The equipped posets of finite representation type were described earlier (in a more general context) in fact in [6], while those of oneparameter type, of tame type and of finite growth in [9], [10], [11] respectively. It was elaborated in [10] a system of differentiation functors (reduction algorithms) for solving these problems, among them the Differentiation VII algorithm and the operation of Completion.
It becomes clear now that, the restriction to the case (R,C) is not essential and the representation theory of equipped posets can be extended (true, not quite automatically) to the case of an arbitrary quadratic field extensionF ⊂G.
At the same time, some other matrix problems of mixed type over the pair (F, G) naturally appear.
In the present paper, we introduce and investigate corepresentations of equipped posets over the pair (F, G) which are in some sense dual to the mentioned above representations. Their classification leads to the dual ma trix problem of mixed type over (F, G). In spite of this, it is not yet known any formal construction establishing a direct relationship between representations and corepresentations.
We define in Section 2 the category of corepresentations corepP of an equipped poset P and develop for it in Sections 3–5 the reduction functor machinery sufficient at least for the finite and oneparameter cases. More pre cisely, the dual variants of Differentiation VII and Completion for corepresen tations are described and (following the scheme of [12] for ordinary posets) the short generalized versions of the algorithms VII anddVII for representations and corepresentations of equipped posets with additional relations of lattice type are constructed.
Possible applications are observed in Section 7.
2. Main definitions and notations
Equipped posets and their representations over the pair of fields of real and complex numbers (R,C) were introduced and studied in [9, 10, 11]. Now we see that the chosen there definition of an equipped poset can be reformulated in the following equivalent form.
A poset (P,≤) is calledequipped if all the order relations between its points x≤y are separated intostrong (xEy) andweak (x¹y) in such a way that
x≤yEz or xEy≤z implies x£z , (2.1) i.e. a composition of a strong relation with any other relation is strong^{∗}.
Clearly, in general both binary relations E and ¹ are not order relations:
they are antisymmetric but not reflexive and¹is not transitive (meanwhileE is transitive).
∗It is interesting that this definition coincides with that one for abiordered set, given in [1] in another context (our relationxEycorresponds toxCyin [1]).
A pointx∈Pwill be calledstrong (weak) ifxEx(resp. x¹x), with the notation in diagrams ◦ (resp. ⊗) (remark that in [9, 10, 11] strong and weak points were calledsingleanddouble respectively). If there are no weak points, the equipment istrivial and the posetPisordinary.
We writex≺yifx¹yandx6=y (similarly,x¢y meansx£yandx6=y).
We call an abstract relationxRy between two pointsstrict ifx6=y.
Remark 2.1. It follows from the definition (2.1) that, for any weak relation x≺y, both the pointsx, y are weak and moreover it holdsx≺t≺y for any possible intermediate point x < t < y. This condition defines completely P (that was the original definition of an equipped poset used in [9, 10, 11]).
By a sumX_{1}+· · ·+X_{n}we denote a disjoint union of subsetsX_{1}, . . . , X_{n} ⊂P (notice that elements belonging to different subsetsXi can be comparable).
For a pointx∈P, set
x^{∨}={y : x≤y}, x^{O}={y : xEy}, x^{g}={y : x¹y},
and dually define subsetsx∧, xM, xf. Remark thatx^{∨}andx^{O} (x∧ andxM) are upper (lower) cones in P, while x^{g} and xf in general are not cones (see the example below). Obviouslyx^{∨}=x^{O}+x^{g}and the dual formula holds . Also it holdsx^{g}=xf=∅for a strong pointx.
For a subsetX ⊂P, set X^{∨}= [
x∈X
x^{∨}, X^{O}= [
x∈X
x^{O}, X^{g} = [
x∈X
x^{g}
and symmetrically (also by the union) define the corresponding setsX∧, XM, Xf. Sometimes we identify a onepoint subset {x} ⊂P with the point itself {x}=x.
Graphically each equipped poset is presented by its Hasse diagram (with strong and weak points) completed by additional lines symbolizing those strong strict relations between weak points which are not consequences of other rela tions.
Example 2.2. If an equipped posetPis given by the diagram
⊗
⊗
⊗
⊗
⊗
⊗
◦
◦
´´´´
´´´´
1 2 3
4 5 6
7 8
then (among strict relations) the relations 1¢{3,5,6}, 4¢8¢6 and 7¢8 are the only strong ones, hence all those in the rest are weak. In particular 3M={1},3f={2,3,4,5},3^{O}=∅,3^{g}={3},1^{O}={3,5,6},1^{g}={1,2},1M=
∅,1f={1},8^{O}={6,8},8M={4,7,8},8f= 8^{g}=∅,7^{O}={6,7,8},7M={7}.
Let F ⊂ G be an arbitrary quadratic field extension and G = F(u) for some fixed elementu. Then each elementx∈Gis presented uniquely in the form α+βuwith α, β∈F and (analogously to the case (F, G) = (R,C)) the coefficientsαandβ are called thereal andimaginary parts ofx.
Each equipped poset P naturally defines a matrix problem of mixed type over the pair (F, G). Consider a rectangular matrixM separated into vertical stripes Mx, x ∈ P, with Mx being over F (over G) if the point x is strong (weak):
M =
x y
. . . Mx . . . My . . . (2.2) Such partitioned matrices M are called matrix representations of P over (F, G). Their admissible transformationsare as follows:
(a) Felementary row transformations of the whole matrixM;
(b) Felementary (Gelementary) column transformations of a stripeMx
if the pointxis strong (weak);
(c) In the case of a weak relationx≺y, additions of columns of the stripe M_{x} to the columns of the stripeM_{y} with coefficients inG;
(d) In the case of a strong relationx¢y, independent additions both real and imaginary parts of columns of the stripeMxto real and imaginary parts (in any combinations) of columns of the stripe My with coeffi cients inF (assuming that, for y strong, there are no additions to the zero imaginary part ofMy).
Two representations are said to be equivalent or isomorphic if they can be turned into each other with help of the admissible transformations. The corresponding matrix problem of mixed type over the pair (F, G) consists of classifying the indecomposable in the natural sense matricesM, up to equiva lence.
One can give another natural definition of representations, in terms of vector spaces overF andG. Identifying the direct sumU_{0}^{2}of two copies of anFspace U0with aGspaceU_{0}^{2}=U0⊕uU0, we notice that, for eachGsubspaceX ⊂U_{0}^{2}, its real and imaginary parts coincide ReX = ImX. Hence X is contained in itsFhull F(X) = (ReX)^{2}, which is aGsubspace inU_{0}^{2}.
A Gsubspace X ⊂ U_{0}^{2} is called strong if F(X) = X or, equivalently, if X =Y^{2} for someFsubspace Y (remark that we use the notation X ⊂Y for an arbitrary inclusion of sets, not necessarily proper).
Arepresentationof an equipped posetPover the pair (F, G) is any collection of the form
U = (U0, Ux : x∈P) (2.3)
where U0 is a finitedimensional Fspace andUx areGsubspaces inU_{0}^{2} such that the following conditions are satisfied
x≤y=⇒Ux⊂Uy,
x£y=⇒F(Ux)⊂Uy. (2.4) In particular, if a point x is strong, then the corresponding subspace Ux = F(Ux) = (ReUx)^{2} is strong and is determined completely by its real part ReUx.
Representations are the objects of the category repP, morphisms U −→^{ϕ} V of which are Flinear maps ϕ : U0 → V0 such that ϕ^{2}(Ux) ⊂ Vx for each x ∈ P. Two representations U, V are isomorphic (U ' V) if and only if for someFisomorphismϕ:U0→V0it holdsϕ^{2}(Ux) =Vxfor allx.^{†}
The described problem (for the classical pair of fields (R,C)) was investigated in [9, 10, 11], where in particular the criteria for an equipped poset to be respectively oneparameter, tame and of finite growth were obtained (the finite representation type criterion for an arbitrary pair (F, G) follows from the earlier result of [6] on schurian vector space categories).
The aim of the present article is to observe one more matrix problem of mixed type over the pair (F, G), which also is defined by an equipped posetP and is in some intuitive sense dual to the mentioned above. Moreover, as will be shown, one can manage with that problem by similar technical means.
Nevertheless, it is not yet known any formal construction reducing one of these problems to another one. Our intension is in particular to present some primary information and facts concerning the new one, on the base of the expe rience with the old one. To avoid confusions in terminology, we will attach the particleco to some terms concerning the new dual problem (saying coproblem, corepresentation, etc.).
Consider again a rectangular separated matrixM of the form (2.2) suppos ing now that all its vertical stripesMx, x∈P, are overG. This is by defini tion amatrix corepresentation of an equipped poset Pover the pair (F, G) to which one can apply the followingadmissible transformations (compare with the transformations (a)−(d) above):
(a^{0}) Gelementary row transformations of the whole matrixM;
(b^{0}) Gelementary (Felementary) column transformations of a stripeMx
if the pointxis strong (weak);
†The matrix problem for representations (a)−(d) naturally appears when classifying the objects of the category repP, up to isomorphism. For this, one should attach to a representationU its matrix realizationM = (Mx : x∈P) in the following way. If a point xis strong (weak), then the columns of the stripeMx are formed by the coordinates (with respect to some base ofU0) of a system of generators of theFspace ReUx(resp. Gspace Ux) modulo its radical subspace ReUx (resp. Ux) defined analogously to [11], Section 3.
Changing the base and the systems of generators, you get the problem (a)−(d).
(c^{0}) In the case of a weak relationx≺y, additions of columns of the stripe Mx to the columns of the stripeMy with coefficients inF;
(d^{0}) In the case of a strong relationx¢y, additions of columns of the stripe Mx to the columns of the stripeMy with coefficients inG;
Thematrix coproblem over the pair (F, G), defined by an equipped posetP, consists in classifying all indecomposable corepresentations, up to equivalence with respect to the admissible transformations (a^{0})−(d^{0}). In this situation also is possible to give a nice invariant definition in terms of subspaces overF and G.
Assume now that U0 is a Gspace. Then for any Fsubspace X ⊂U0 it is defined its Ghull G(X) = GX being nothing else but the ordinary Gspan of X, i.e. the minimal Gsubspace in U0 containing X. If G(X) = X, the FsubspaceX itself is aGspace and is said to beGstrong.
Acorepresentationof an equipped posetPover the pair (F, G) is a collection of the form
U = (U0, Ux : x∈P) (2.5)
whereU0 is a finitedimensionalGspace containing Fsubspaces Uxsuch that x≤y=⇒Ux⊂Uy,
x£y=⇒G(Ux)⊂Uy. (2.6)
Notice that to a strong pointxa strong subspaceUx=G(Ux) corresponds.
Corepresentations are the objects of the category corepP, with morphisms U −→^{ϕ} V being Glinear maps ϕ : U0 → V0 such that ϕ(Ux) ⊂ Vx for each x∈P. It is clear that two corepresentationsU, V areisomorphic if and only if for someGisomorphismϕ:U_{0}→V_{0}it holdsϕ(U_{x}) =V_{x} for allx.
Remark 2.3. The classification of indecomposable objects of the category corepP, up to isomorphism, corresponds precisely to the described above ma trix coproblem (a^{0})−(d^{0}) (if to exclude from considerations formal indecom posable “empty” matrices having zero rows and one column). Namely, ifM is a matrix corepresentation, one may attach tonrows ofM a basee1, . . . , en of some ndimensionalGspace U0 and identify each column (λ1, . . . , λn)^{T} ofM with the element u=λ1e1+· · ·+λnen ∈U0. Denoting then byF[X] (resp.
G[X]) theFspan (Gspan) inU0of any column setX ⊂M, put Ux=X
y¹x
F[My] +X
y£x
G[My]
and obtain immediately a collection (2.5) satisfying the conditions (2.6). It is clear that each vertical stripe Mx represents (by its columns) a system of generators of the spaceUx modulo itsradical subspace
Ux=X
y≺x
F[My] +X
y¢x
G[My],
hence the transformations (a^{0})−(d^{0}) ofM reflect both base changing inU0and generator changing in subspacesUx.
3. Further notations and preliminaries
The dimension of a matrix corepresentation M is a vector d = dimM = (d_{0}, d_{x} : x∈P) withd_{0}(resp. d_{x}) being the number of rows inM (of columns inMx). Meanwhile thedimension ofU is a vectord= dimU = (d0, dx : x∈P) with d0 = dimGU0 and dx = dimFUx/Ux (dx = dimGUx/Ux) for a weak (strong) pointx.
Obviously dimU ≤ dimM (the equality holds if and only if the columns of each stripe Mx are linearly independent modulo the radical columns). A corepresentationU will be calledtrivial if dimGU0= 1.
Asincere vector has no zero coordinates by definitions. A representation or corepresentation is sincere if its dimension vector is sincere. Every equipped poset having at least one sincere indecomposable representation (corepresenta tion) is calledsincere with respect to representations (corepresentations).
A subset ofPis achain(antichain) if all its points are pairwise comparable (incomparable). Thelength of a chain is the number of its points. A chain of the forma1 ≺a2 ≺ · · · ≺an is calledweak, if additionallya1 ≺an then it is completely weak.
An arbitrary subsetX ⊂Pis said to becompletely weakif all its points and possible relations between them are weak.
For a subsetX ⊂Pand a matrix representation or corepresentationM, set MX =S
x∈XMx.
Denote by minX (maxX) the set of all minimal (maximal) points of a subset X ⊂P.
Let (X, Y) be any pair of subsets of Psuch thatX is completely weak,Y is arbitrary and X ∩Y^{∨} = ∅. We use in the sequel trivial indecomposable corepresentationsTb(X, Y) of the form
Tb(X, Y) = (G, Ut : t∈P) where
Ut=
G, ift∈X^{O}∪Y^{∨}; F, ift∈X^{∨}\(X^{O}∪Y^{∨});
0, otherwise.
It is clear that Tb(X, Y) =T(minb X,minY), thus in principle one can deal with objectsTb(X, Y) supposingX, Y to be antichains.
Setting
Tb(X,∅) =T(Xb ), Tb(∅, Y) =P(Yb ), we have in particular
Tb(∅,∅) =T(∅) =b P(∅) = (G,b 0, . . . ,0).
In the case X = {x} or Y = {y}, we write simply Tb(x, Y) or Tb(X, y), so for instance, the objectsTb(x),Tb(x, y) withx≺y andPb(y) are partial cases of Tb(X, Y) and have the following matrix forms
T(x) =b x
1 , T(x, y) =b
x≺y
1 u , P(y) =b y 1 u
(clearly, the element u can be deleted from the matrix P(y) if the pointb y is strong). In Section 5, some other objects of type T(x, Yb ) will be considered (see Theorem 5.7).
The following simple fact holds.
Lemma 3.1. The corepresentationsPb(∅),Pb(x),Tb(x)andTb(x, y)are all possi ble (up to isomorphism) indecomposable corepresentations of an arbitrary weak chain.
Sketch of the proof. Use induction on n. The case n= 1 is in fact trivial. If n ≥2, set X ={x2, . . . , xn} and consider a matrix corepresentation M of a weak chainx1≺ · · · ≺xn. First reduce the stripeMx1 to the natural canonical form, with direct summands Pb(x1),Tb(x1) and some zerorows. Since each direct summandP(xb 1) annuls (by admissible column additions Mx1 −→MX) the same row in MX and is in fact a direct summand of the whole matrix M, one can assumeMx1 containing (besides zerorows) only direct summands Tb(x1). ThusM takes the form
M =
x1 x2 . . . xn
I ∗ ∗ ∗ K
0 ∗ ∗ ∗ L (3.1)
Now you can reduce, by induction step, the stripe MX ∩L to the canonical form with direct summands mentioned in Lemma and then finally reduce (using admissible column additions Mx1 −→ MX and row additions L −→^{G} K) the stripeMX∩K getting the desired result (some more proof details for the case (R,C) are given in [8]). ¤X We recall (see [10, 12]) that, for given subspacesA, B, X, Y of some vector space V over a field, the pair (X, Y) is called (A, B)cleavingifV =X⊕Y and A=X+ (A∩B), B=Y ∩(A+B).
Denote byU^{m} (resp. ϕ^{m}) the direct sum of mcopies of a representation, corepresentation or a spaceU (of a morphism or linear map ϕ).
IfX is a set and U a vector space, then U^{X} means the direct some ofX
copies ofU numbered by the elements ofX.
In the sequel,K{e1, . . . , en}is a notation for the vector space over a fieldK generated by the given vectorse1, . . . , en.
By [U] we denote the isomorphism class of an objectU. For a collection of object X, set [X] ={[U] : U ∈X}. Let IndP(resp. dIndP) be the set of all isomorphism classes of indecomposable objects in repP(corepP).
Sometimes (if no confusions) a onepoint set{a} is denoted simply bya.
4. Differentiation VIId
The combinatorial action of DifferentiationVII coincides with that one of Difd ferentiation VII described in [10]. Namely, a pair of incomparable points (a, b) of an equipped poset Pis called VIIsuitable or dVIIsuitable ifais weak, b is strong and
P=a^{O}+bM+{a≺c1≺ · · · ≺cn}
where {a≺c1≺ · · · ≺cn} is a completely weak chain incomparable with the point b. Putting a =c0, we assumen ≥0. DenoteA =a^{O}, B = bM\b and C={c1≺ · · · ≺cn}.
Thederived posetP^{0} =P^{0}_{(a,b)}ofPwith respect to such a pair (a, b) has the form
P^{0}_{(a,b)}= (P\(a+C)) +{a^{−} < a^{+}}+C^{+}+C^{−}
where the point a^{−} is weak, the point a^{+} is strong, C^{−} = {c^{−}_{1} ≺ · · · ≺c^{−}_{n}} and C^{+} = {c^{+}_{1} ≺ · · · ≺ c^{+}_{n}} are completely weak chains, c^{−}_{i} ≺ c^{+}_{i} for all i= 1, . . . , n; a^{−}≺c^{−}_{1}; a^{+}< c^{+}_{1}; c^{−}_{n} < band the following natural conditions are satisfied:
(a) Each of the pointsa^{−}, a^{+}(c^{−}_{i} , c^{+}_{i} ) inherits all previous order relations of the original pointa(ci) with the points of the subsetP\(a+C).
(b) The order relation inP^{0}_{(a,b)} is induced by the initial order relation in the subsetP\(a+C) and by the listed above relations.
A
a c1
cn b
B
⊗
⊗
⊗
⊗
¡¡
¡¡
¡¡ µ´
¶³
◦
µ´
¶³
V IId

(a, b)
A
a^{+} c^{+}_{1}
c^{+}_{n}
a^{−} c^{−}_{1}
c^{−}_{n} b
B
◦ µ´
¶³
⊗
⊗
⊗
¡¡
¡¡
¡¡
@@
@@
@@
⊗
⊗
⊗
¡¡
¡¡
¡¡
⊗¡¡
@@
◦
µ´
¶³
Thedifferentiation functor Db_{(a,b)}: corepP−→corepP^{0} (denoted also by^{0}) of the algorithmdVII assigns to each corepresentationU ofPthederived oneU^{0}
ofP^{0} by the rule
U_{0}^{0} =U0
U_{c}^{0}−
i =Uci∩Ub fori= 0,1, . . . , n U_{c}^{0}+
i =Uci+G(Ua) fori= 0,1, . . . , n U_{x}^{0} =Ux for the remaining pointsx∈P^{0}.
(4.1)
And, for a morphism U −→^{ϕ} V of the category corepP(considered as a linear map ϕ:U0 −→V0), set ϕ^{0} =ϕ. One checks trivially that the functor is well defined.
The objectsPb(a), Tb(a) andTb(a, ci), i= 1, . . . , n, play an important role in the description of properties of the algorithmVII. Their derivative all coind cidePb(a)^{0} =Tb(a)^{0} =Tb(a, ci)^{0} =Pb(a^{+}), thus we have to consider the reduced objects of the category corepP (corepP^{0}) as those not containing direct sum mandsPb(a), Tb(a) andT(a, cb _{i}), i= 1, . . . , n, (resp. Pb(a^{+})).
We point out that, as a rule, the derived object U^{0} contains trivial direct summandsPb(a^{+}), even ifU is indecomposable. That’s why thereduced derived object U^{↓} (which is unique up to isomorphism) is defined for any objectU ∈ corepP^{0} as the largest direct summand ofU^{0} not containing trivial summands Pb(a^{+}), i.e. by setting U^{0} =U^{↓}⊕Pb^{m}(a^{+}), with m= dimGG(Ua)/(G(Ua)∩ Ub) = dimG(G(Ua) +Ub)/Ub. Evidently (U1⊕U2)^{↓}'U_{1}^{↓}⊕U_{2}^{↓}.
An equivalent definition of U^{↓} is as follows: take any (G(Ua), Ub)cleaving pair of subspaces (E0, W0) of theGspaceU0and setU^{↓}=W = (W0;Wxx∈ P^{0}) whereWx=U_{x}^{0} ∩W0 for eachx∈P^{0}.
It is clear that, the reduced derived objectU^{↓} can be viewed as a corepre sentation not only of P^{0}_{(a,b)} but also of thecompleted derived equipped poset P^{0}_{(a,b)}obtained fromP^{0}_{(a,b)}by adding one additional relationa^{+}< b(since due to the definitionW_{a}^{+}⊂Wb).
The integration procedure for the algorithm dVII (which is in some sense inverse to the differentiation) is described in the following way. For a given corepresentation W of the completed derived poset P^{0}_{(a,b)}, present each F spaceW_{c}^{+}
i (i= 1, . . . , n) in the formW_{c}^{+}
i =W_{c}^{+}
i ⊕Si⊕Hi, whereSi, Hi are some complements such that Si ⊂Wb and Hi∩Wb = 0. Choose in each Si
some Fbasesi1, . . . , simi. Analogously present theGspaceW_{a}^{+} in the form W_{a}^{+} = W_{a}+⊕T0 where T0 = G{t01, . . . , t0m0} is some complement for the GsubspaceW_{a}+.
Taking now a newGspaceE0with a base{eij : i= 0, . . . , n;j = 1, . . . , mi}, attach toW itsprimitive objectW^{↑}=U = (U0, Ux : x∈P) where
U0=W0⊕E0,
Ux=Wx⊕E^{A∩{x}}_{0} for x6=a, ci ,
Ua=W_{a}−+F{s0j+ue0j : j= 1, . . . , m0}+
+F{eij : i= 0, . . . , n;j= 1, . . . , mi}, Uci =Uci−1+W_{c}^{−}
i +Hi+F{sij+ueij : j= 1, . . . , mi} (i≥1), (4.2)
and Uc0 = Ua. The primitive object W^{↑} depends no, up to isomorphism, on the choice of subspaces Ti, Hi and their bases, moreover (W1⊕W2)^{↑} ' W_{1}^{↑} ⊕W_{2}^{↑}. There hold also equalities dimGS0 = dimG(Wa^{+}/G(Wa^{−})) and dimFSi = dimF(W_{c}^{+}
i ∩Wb)/(W_{c}^{−}
i +W_{c}^{+}
i−1∩Wb) fori= 1, . . . , n.
The main result on DifferentiationdVII is as follows.
Theorem 4.1. In the case of DifferentiationdVII, the operations↑and↓induce mutually inverse bijections
dIndP\ [P(a),b T(a),b Tb(a, ci), i= 1, . . . , n] ¿ dIndP^{0}_{(a,b)}=dIndP^{0}_{(a,b)}\[Pb(a^{+})]. Proof. For given corepresentations U of P and W of P^{0}, one has to prove that [U^{↓}]^{↑} 'U and [W^{↑}]^{↓}'W. The second isomorphism is verified without difficulties by a standard routine procedure using the formulas (4.1) and (4.2), this is left to the reader as an exercise.
To prove the first one, consider the matrixM of a reduced corepresentation U ofPchosen in such a way that the columns of each vertical stripeMx,x∈P, generateUx. Applying toM suitableGelementary row transformations, place at its bottom linearly independent rows corresponding to some base of the G subspaceUbobtaining all zeroes above them in the blockMb+B(our convention for matrix pictures is that empty blocks denote zeroblocks, but blocks marked by∗are arbitrary, andIdenotes the identity block of arbitrary order):
a A c1 cn B b
uI I
I uI
I uI
∗ X1 Xn
∗ ∗ ∗ ∗ S1 Y1 ∗ Sn Yn ∗ ∗
a^{−} a^{+} c^{−}_{1} c^{−}_{n}
E0
W0
(
Q Ub
 {z } c^{+}_{1}
 {z } c^{+}_{n}
(4.3) Further, select linearly independent over G rows in Ma above the horizontal stripeUb denoting the new horizontal stripe by E0and obtaining (by suitable
Gelementary transformations of rows) all zeroes in the intermediate horizontal stripeQ∩Ma. Applying then to the blockQ∩(Mc1∪· · ·∪Mcn) suitable admis sible column transformations, we can leave there only those cells X1, . . . , Xn
the columns of which areFlinearly independent (all together).
Consider each matrix Mci (i = 1, . . . , n) as a union of two vertical stripes Mci = M_{c}^{0}_{i}∪M_{c}^{00}_{i} where M_{c}^{00}_{i} is formed by the columns containing the block Xi and M_{c}^{0}_{i} consists of the rest of the columns. Reduce to the canonical form the block E0∩(Ma∪M_{c}^{0}_{1}· · · ∪M_{c}^{0}_{n}) considering it as a corepresentation of the completely weak chaina≺c1≺ · · · ≺cn and applying Lemma 3.1 (select the matrix forms T(a) =b u 1 andTb(a, ci) = 1 u , i= 1, . . . , n). Omit the direct summandsPb(a) which obviously are split as direct summands of the whole M. Make (by row additions) zeroes below the identity blocks I in M_{a} and get the blockMa as shown in (4.3).
Then (using row additions Q −→^{G} E0 and column addition Ma F
−→
M{c1,...,cn}) annul all the blocksM_{c}^{00}_{i} (this is possible because the matrixX1∪
· · · ∪Xn can be viewed as a corepresentation of a completely weak chain c1≺ · · · ≺cn and hence presented as a direct sum of the mentioned in Lemma 3.1 trivial blocks). Remark that the shown matrix blocks S_{i} and X_{i} ∪Y_{i} correspond to the subspaces Si and Hi respectively, considered above in the integration procedure.
Annul finally (by column additionsMa G
−→MA) the blockE0∩Aand obtain the shown matrix form (4.3). An immediate comparison with the formulas (4.1) and (4.2) confirms evidently that in the horizontal stripeW0=Q∪Ubwe have just the reduced derived corepresentation U^{↓} and certainly the isomorphism
(U^{↓})^{↑}'U holds. The proof is complete. ¤X
Remark 4.2. Analogously to the case of Differentiation VII (see [10], Remark 3.5), one can expect that the differentiation functordVII induces an equivalence of the quotient categories
corepP/hPb(a),Tb(a),Tb(a, ci), i= 1, . . . , ni^{→}^{−}^{∼} corepP^{0}/hP(ab ^{+})i
where the brackets h. . .i denote the ideals of all morphisms passed through finite direct sums of the shown objects. The proof may be carried out by the same scheme as in [12] (Section 7) using short generalized steps of the algorithm dVII, as explained below.
5. Short generalized versions of Differentiations VII and VIId
To deal with differentiation algorithms more effectively, one has to reduce (if possible) long differentiation steps to shorter ones, probably passing to a more wide (but suitable) class of matrix problems.
Such a possibility exists in the case of the algorithms VII and VII andd can be realized analogously to [12] via passing to a class of representations or corepresentations of equipped posets with additional relations. We outline here briefly the main scheme (a more detailed exposition will be placed elsewhere).
First we decompose easily the algorithms VII and dVII in two intermediate steps, one of which will be decomposed then more.
Preliminary decomposition. Let (a, b) be a VIIsuitable pair of points of an equipped poset P as defined above, i.e. P = a^{O}+bM+ (a+C) where C={c1≺ · · · ≺cn} is a completely weak chain (n≥0, c0=a) incomparable withb(set A=a^{O}, B=bM\b).
We recall that the combinatorial action of the algorithms VII anddVII coin cide and are going to present it as a combination of two steps.
Thelong step or brieflylstep or Differentiation VIIl consists in transition P 7→ P˙(a,b) from the ordinary equipped poset P to an equipped poset with relation of the form
P˙(a,b)= (P_{(a,b)}^{l} Σ(a,b)),
where P_{(a,b)}^{l} is a new ordinary equipped poset differing slightly from P^{0}_{(a,b)}, namely
P_{(a,b)}^{l} = (P\C) +C^{+}+C^{−}
whereC^{−}={c^{−}_{1} ≺ · · · ≺c^{−}_{n}}andC^{+}={c^{+}_{1} ≺ · · · ≺c^{+}_{n}} are completely weak chains,c^{−}_{i} ≺c^{+}_{i} for i= 1, . . . , n; a¢c^{+}_{1}; c^{−}_{n} < b and the standard conditions hold:
(al) Each of the points c^{−}_{i} , c^{+}_{i} ,(i= 0,1, . . . , n) inherits all previous order relations of the original pointciwith the points of the subsetP\(a+C).
(bl) The order relation inP_{(a,b)}^{l} is induced by the initial order relation in the subsetP\Cand by the listed above relations.
As for the set of relations Σ_{(a,b)}, it consists of one relation only Σ_{(a,b)}={ab⊂c^{−}_{1} }
which means conditionally that the categories rep ˙P(a,b) and corep ˙P(a,b) are by definition the full subcategories of the categories repP_{(a,b)}^{l} and corepP_{(a,b)}^{l} respectively formed by all those objectsV which satisfy the relations
Va∩Vb⊂V_{c}^{−}
1 . The diagram of the long step is as follows:
A B
a c1
cn b
⊗
⊗
⊗
⊗
¡¡
¡¡
¡¡ µ´
¶³
◦
µ´
¶³ ^{V II}^{l}
(a, b) A B
a c^{−}_{1}
c^{+}_{1} c^{−}_{n} c^{+}_{n}
b
µ´
¶³
⊗
⊗
⊗
⊗
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¡¡
¡¡
¡¡
@@
@@
@@
⊗
⊗
⊗
¡¡
¡¡
¡¡◦ µ´
¶³
ab⊂c^{−}_{1}
P P˙(a,b)
Notice that if n = 0 and therefore C = ∅, then Σ(a,b) = ∅ and actually P˙(a,b)=P_{(a,b)}^{l} =P.
The additional0step or Differentiation VII0 is a transition from ˙P(a,b) to P^{0}_{(a,b)} where P^{0}_{(a,b)} is the defined in Section 4 complete (a, b)derived poset.
In other words, Differentiation VII0 is nothing else but a particular case of Differentiation VII applied in the situationC=∅:
A
B a
b
⊗ µ´
¶³ ◦
µ´
¶³
V II_{0}

(a, b)
A
a^{+} a^{−}
b
B
◦ µ´
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⊗¡¡¡¡
@@
◦
µ´
¶³
P P_{(a,b)}^{0}
So, the obtained combinatorial decompositionP 7−→P˙_{(a,b)} 7−→ P^{0}_{(a,b)}(com pare with the shown in Section 4 diagram of the algorithm VII) correspondsd to the functor decomposition for representations and corepresentations
D(a,b)=D_{(a,b)}^{0} D_{(a,b)}^{l} and Db(a,b)=Db_{(a,b)}^{0} Db_{(a,b)}^{l} with the functors being defined in the following unified way.
For a representation or corepresentationU ofP and a pointx∈P, let Ufx
be thehull of the spaceUxin the following sense combining two possibilities Ufx=
½ F(Ux), if U is a representation;
G(Ux), if U is a corepresentation.
Then both the differentiation functorsD_{(a,b)}andDb_{(a,b)}of the algorithms VII and dVII (described in [10] and Section 4 above respectively) are given by the
same formulas
U_{0}^{0} =U0
U_{c}^{0}−
i =Uci∩Ub fori= 0,1, . . . , n U_{c}^{0}+
i =Uci+Ufa fori= 0,1, . . . , n
U_{x}^{0} =Ux for the remaining pointsx∈P^{0}
ϕ^{0}=ϕ for a linear mapmorphismϕ:U0−→V0.
(5.1)
To get from (5.1) thelstep differentiation functorsD_{(a,b)}^{l} andDb_{(a,b)}^{l} , you have simply to exclude the casei= 0. Meanwhile to get the 0step functorsD_{(a,b)}^{0} andDb_{(a,b)}^{0} , you have on the contrary to assumen= 0.
Denote byU^{(l)}(resp. U^{(0)}) the derivative of some object (representation or corepresentation)U with respect to thelstep (0step) of Differentiation. Then it holds evidently for representations
P(a)^{(l)}=P(a), T(a)^{(l)}=T(a, ci)^{(l)}=T(a),
P(a)^{(0)}=P(a^{+}), T(a)^{(0)}=P^{2}(a^{+}) (5.2) and for corepresentations
Tb(a)^{(l)}=Tb(a, ci)^{(l)}=T(a),b Pb(a)^{(l)}=Pb(a),
Tb(a)^{(0)}=Pb(a)^{(0)} =Pb(a^{+}) (5.3) whereP(x), T(x) andT(x, y) are representations in the matrix form
P(x) = x
1 , T(x) = x 1
u , T(x, y) =
x≺y 1 0 u 1 (recall thatPb(x),T(x),b T(x, y) have been already defined in Section 3).b
Taking the equalities (5.2) and (5.3) into account, analogously to Differen tiations VII and dVII, one can define naturally the reduced derived object U^{↓} of an objectU for the algorithms VIIl (resp. dVIIl) as a maximal direct sum mand of U^{(l)} not containing summands T(a) (resp. Tb(a)). Also analogously to the algorithms VII and dVII one should define theprimitive object W^{↑} for each object W of the derived category (free of direct summands T(a) (resp.
Tb(a))) and to deduce then the following main property of thelstep algorithm (compare with Theorem 4.1).
Theorem 5.1. In the case of Differentiations VIIl andVIIdl, the operations ↑ and↓ induce mutually inverse bijections
IndP\ [T(a), T(a, ci), i= 1, . . . , n] ¿ Ind ˙P(a,b)\ [T(a)], dIndP\ [Tb(a),Tb(a, ci), i= 1, . . . , n] ¿ Ind ˙dP(a,b)\ [Tb(a)].
We left the details of the proof (which is very similar to the proofs of Theorem 3.5 in [10] and Theorem 4.1 above^{‡}) as an exercise for the interested reader.
Since the 0step algorithms VII0 anddVII0are special cases of VII and dVII, we obtain for them immediately from Theorem 3.5 in [10] and Theorem 4.1 above the following corollary.
Corollary 5.2. In the case of Differentiations VII0 anddVII0, the operations
↑ and↓ induce mutually inverse bijections
IndP\ [P(a), T(a)] ¿ IndP_{(a,b)}^{0} \ [P(a^{+})], dIndP\ [Pb(a),Tb(a)] ¿ dIndP_{(a,b)}^{0} \ [P(ab ^{+})].
Remark 5.3. In accordance with the previous statements, one should expect that the differentiation functors VIIl, dVIIl, VII0, VIId0 induce respectively equivalences of the quotient categories
(i) repP/hT(a), T(a, ci), i= 1, . . . , ni^{−}^{→}^{∼} rep ˙P_{(a,b)}/hT(a)i, (bi) corepP/hT(a),b T(a, cb i), i= 1, . . . , ni^{−}^{→}^{∼} corep ˙P_{(a,b)}/hT(a)ib , (ii) repP/hP(a), T(a)i^{−}^{→}^{∼} repP_{(a,b)}^{0} /hP(a^{+})i (C=∅), (ii) corepb P/hP(a),b Tb(a)i^{−}^{→}^{∼} corepP_{(a,b)}^{0} /hPb(a^{+})i (C=∅).
Our next goal is to decompose more essentially thelstep.
Main decomposition. First we define the combinatorial action of the short generalized algorithm VIIswith respect to a triple of points.
A triple of points (a, b, c) of an equipped posetPwill be calledVIIssuitable if the pointsa, care weak, bis strong incomparable witha, cand
P=a^{O}+bM+{a≺X ≺c≺Y}
where{a≺X ≺c≺Y} is a completely weak set containing arbitrary subsets X, Y (probably empty).
Thederived or (a, b, c)derived equipped posetwith relations P^{0}_{(a,b,c)}of the posetPis a pair
P^{0}_{(a,b,c)}= (P_{(a,b,c)}^{s} Σ_{(a,b,c)}) where
P_{(a,b,c)}^{s} = (P\c) +{c^{−}, c^{+}}
is an equipped poset such that the pairs c^{−} ≺ c^{+}, X ≺ c^{+} and c^{−} ≺Y are completely weak, a¢c^{+}, c^{−} < b and the partial order in P_{(a,b,c)}^{s} is induced both by these relations and by the initial order inP\c(it is assumed that each of the points c^{−}, c^{+} inherits the order relations of the pointc with the points of the subseta^{O}+bM).
‡The only difference with the complete matrix Differentiations VII anddVII is that one needs no more to separate the blockMainto the parts corresponding to the pointsa^{±}.
Further, Σ_{(a,b,c)}is a set of two formal relations Σ_{(a,b,c)}={c^{+}⊂ea+^{_}Y; bX⊂c^{−}},
and by definition the categories repP^{0}_{(a,b,c)} and corepP^{0}_{(a,b,c)} are the full sub categories of the categories repP_{(a,b,c)}^{s} and corepP_{(a,b,c)}^{s} respectively formed by those objectsW which satisfy the relations
Wc^{+} ⊂fWa+W^{_}Y and Wb∩WX⊂Wc^{−}
where W^{_}Y =T
y∈Y Wy and WX = P
x∈XWx (with the commonly accepted convention thatW^{_}∅ =W0 andW∅ = 0).
A X
Y
B
a
c
b
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µ´
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µ´
¶³
⊗
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µ´
¶³ 0
(a, b, c) A
X
Y
B
a
c^{+}
c^{−}
b
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c^{+}⊂ea+^{_}Y; b(a+X)⊂c^{−}
P P^{0}_{(a,b,c)}
Then both the differentiation functors D(a,b,c) : repP −→ repP^{0}_{(a,b,c)} and Db_{(a,b,c)}: corepP−→corepP^{0}_{(a,b,c)}(also denoted briefly by^{0}) are given by the same formulas
U_{0}^{0} =U0
U_{c}^{0}+=Uc+Ufa; U_{c}^{0}− =Uc∩Ub
U_{x}^{0} =Ux for the remaining points x∈P^{0}_{(a,b,c)} ϕ^{0}=ϕ for a linear mapmorphismϕ:U0−→V0.
(5.4)
Remark 5.4. Certainly, the action of the functor is naturally extended to those situations when the initial poset P itself is an equipped poset with relations.
In such cases some more relations have to be added to Σ_{(a,b,c)}.
Proposition 5.5. Let P=a^{O}+bM+{a≺c1≺ · · · ≺cn} be an equipped poset with VIIsuitable pair of points (a, b). Then the long differentiation functors D_{(a,b)}^{l} andDb_{(a,b)}^{l} of the algorithmsV IIlanddVIIlare presented as compositions
D_{(a,b)}^{l} =D_{(a,b,c}_{1}_{)}D_{(a,b,c}_{2}_{)}. . . D_{(a,b,c}_{n}_{)}, Db_{(a,b)}^{l} =Db(a,b,c1)Db(a,b,c2). . .Db(a,b,cn)