c Birkh¨auser Verlag, Basel, 2001
Annals of Combinatorics 5 (2001) 459-478
Annals of Combinatorics
0218-0006/01/040459-20$1.50+0.20/0
Directed Graphs and the Combinatorics of the Polynomial Representations of GLn (C) Miguel A. M´endez IVIC, Departamento de Matem´atica, and UCV, Facultad de Ciencias, Departamento de Matem´atica, Caracas, Venezuela
[email protected] Received December 5, 2000 AMS Subject Classification: 05E15
Dedicated to the memory of Gian-Carlo Rota Abstract. Using dierected graphs, we present a combinatorial model for the polynomial matrices corresponding to representations of the general linear groups. In doing so, we obtain a very simple combinatorial rule to multiply basic elements of the Schur algebra. Keywords: Schur algebras, MacMahon master theorem, combinatorial species
1. Introduction This work was inspired in problems posed in previous articles by Louck [10] and Chen and Louck [3], where many connections of the MacMahon master theorem and combinatorics with the representations of GLn (C) and U(n) were pointed out. The general problem addressed in [3] was that of combinatorially describing the matrices Dλ (X), λ a partition of k, and X = (xi, j )ni, j=1 , that give the irreducible unitary representations of U(n). They explore the case λ = (k) giving a combinatorial proof of the invariance of D(k) (X); D(k) (XY) = D(k) (X)D(k) (Y). In [8] we introduced the matrix species (see also [9]) as a generalization of the classical Joyal species (see [6], and [1]). This kind of species are useful in dealing with the problem of enumerating structures ‘built’ over arc-labeled digraphs. In this article, by describing in terms matrix species the identity R(XY) = R(X)R(Y), we obtain in Theorem 4.3 and Proposition 5.2 a very simple combinatorial rule to multiply two basic elements of the Schur algebra (see [4, 5, 7]). This product, described some times as ‘clumsy,’ turns out to be a very natural one when reinterpreted in terms of digraphs and Eulerian bijections. Aiming at the general problem posed in [3] we describe combinatorially the matrices E(k) (X) (exterior power) and S(k) (X) (symmetric power) corresponding to the 459
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1 1 d
a
2
b 2
c
a d
1 2
c 3
3
b
3
Figure 1: Directed graph and corresponding bipartite digraph. shapes λ = (k) and λ = (1k ). Using the combinatorial rule obtained in Theorem 4.3 we give purely combinatorial proofs of the invariance of these matrices. 2. Weighted Digraphs and Operations Let V be a finite totally ordered set. An arc-labeled digraph G with arc label set Arc(G) and vertices in V is a triplet (Arc(G), tG , hG ), where Arc(G) is a finite set and tG and hG are functions tG , hG : Arc(G) → V. The components of the ordered pair (tG (e), hG (e)) are respectively the initial (tail) and final (head) end-points of the arc with label e ∈ Arc(G). A digraph G can also be identified with a matrix of pairwise disjoint sets G = (Ei, j )i, j∈V , Ei, j = tG−1 ({i}) ∩ h−1 G ({ j}).
(2.1)
By convenience we usually represent a digraph as a bipartite digraph by splitting into two copies the vertex set, and drawing each arc pointing from left to right (see Figure 1). A weighted digraph Gw is a pair Gw = (G, w) of a digraph G and a function w : Arc(G) → C. For a subset A ⊆ Arc(G) define the inventory |A|w of A by |A|w =
∑ w(a).
a∈A
The weight matrix of (G, w) is defined by |Gw | = (|tG−1 {i} ∩ h−1 G { j}|w )i, j∈V = Ei, j w i, j∈V .
Observe that if we denote by 1 the weight assigning 1 to each arc of G, |G 1 | is the adjacency matrix of G |G1 | = A (G) = (|Ei, j |)i, j∈V .
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461
An isomorphism α : Gw1 1 → Gw2 2 is any adjacency and weight-preserving bijection α : Arc(G1 ) → Arc(G2 ), tG2 (α(e)) = tG1 (e), hG2 (α(e)) = hG1 (e), w2 (α(e)) = w1 (e), for all e ∈ Arc(G1 ). The weighted digraphs with vertices in V , V fixed, and the isomorphisms form a category denoted by GVω . The weighted digraphs with weight 1 and vertices in V form a subcategory of GVω , denoted by GV . Let Gw1 1 and Gw2 2 be two weighted digraphs with vertices on the same set V . The sum Gw = Gw1 1 + Gw2 2 is the weighted digraph formed by the superposition (disjoint union) of the arcs of G1 and G2 and weight ( w1 (a), if a ∈ Arc(G1 ), w(a) = w2 (a), if a ∈ Arc(G2 ) · The product
Gw = Gw1 1 ? Gw2 2
is the weighted digraph whose arcs are the walks of length two, the first arc being in G1 and the second in G2 . The weight of each walk being the product of the respective weights of the steps. Formally, the arc set of Gw = Gw1 1 ? Gw2 2 is Arc(Gw ) = {(e, f ) ∈ Arc(G1 ) × Arc(G2 )|hG1 (e) = tG2 ( f )}, tG1 ?G2 (e, f ) = tG1 (e), hG1 ?G2 (e, f ) = hG2 ( f ), and w(e, f ) = w1 (e) · w2 ( f ). It is easy to verify that |Gw1 1 + Gw2 2 | = |Gw1 1 | + |Gw2 2 |, |G1w1 ? Gw2 2 | = |Gw1 1 | · |Gw2 2 |, where the product · on the right is matrix product. Consider now two weighted digraphs Gw1 1 and Gw2 2 with possible different sets of vertices, say respectively V1 and V2 . The tensor product Gw1 1 ⊗ Gw2 2 is the weighted digraph with arc set Arc(G1 ⊗ G2 ) = Arc(G1 ) × Arc(G2 ) and vertices in the set V1 ×V2 . The arc (e, f ) goes from (i, r) ∈ V1 × V2 to ( j, s) ∈ V1 × V2 if e goes from i to j in G1 and f goes from r to s in G2 . The weight of an arc (e, f ) ∈ Arc(G1 ) × Arc(G2 ) is w1 (e)w2 ( f ). In symbols Gw1 1 ⊗ Gw2 2 = (Arc(G1 ) × Arc(G2 ), tG1 × tG2 , hG1 × hG2 , w1 · w2 ). It is also easy to verify that |Gw1 1 ⊗ Gw2 2 | = |Gw1 1 | ⊗ |Gw2 2 |, where the product on the right is tensor product of matrices. We have the following formula (Gw1 1 ⊗ Gw2 2 ) ? (Gw3 3 ⊗ Gw4 4 ) ≈ (Gw1 1 ? Gw3 3 ) ⊗ (Gw2 2 ? Gw4 4 ), where “≈” means isomorphism of weighted digraphs.
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3. Weighted Matrix Species Definition 3.1. Let V be a finite totally ordered set. A (weighted, n × n-to-V × V ) matrix species is any covariant functor R from the category of digraphs with vertex set (ω) [n] = {1, 2, . . . , n}, Gn , to the category of weighted digraphs GV . Two digraphs in Gn are isomorphic iff they have the same adjacency matrix. By functoriallity, for a digraph G, the matrix RG := |R[G]| depends only on the adjacency matrix A of G. We denote it by RA . The generating function of R is defined by R(X) = ∑ RA A
XA , A!
where X = (xi, j )ni, j=1 , A = (ai, j )ni, j=1 ranges in the set of n × n matrices with nona negative integer entries, XA = ∏ni, j=1 xi,i,j j , and A! = ∏ni, j=1 ai, j !. Definition 3.2. A matrix species R : Gn → GVw is said to be polynomial if the entries of its generating function are polynomials, i.e., RA = O except for a finite number of A’s. R is said to be of degree k if each entry of its generating function is a homogeneous polynomial of degree k. A polynomial matrix species R is said to be invariant if its generating function satisfies R(In ) = IV ,
(3.1)
R(X · Y) = R(X)R(Y),
(3.2)
where IV = (δi, j )i, j∈V is the V ×V identity matrix, and Y = (yi, j )ni, j=1 . 3.1. Operations (ω)
Let R and S be two matrix species, R, S : Gn → GV . The sum and product of R and S are defined respectively by (R + S)[G] = R[G] + S[G], (R · S)[G] =
∑
R[G1 ] ? S[G2 ].
(3.3) (3.4)
G1 +G2 =G
For the next operation we do not require the codomain of R and S to be the same. (ω) (ω) Assume now that S is a matrix species with codomain GW , S : Gn → GW . Then, (ω) R ⊗ S : Gn → GV ×W is the matrix species defined by the formula (R ⊗ S)[G] =
∑
R[G1 ] ⊗ S[G2]·
G1 +G2 =G
Remark 3.3. The total order in V × W is the lexicographic order with respect to the orders in V and W .
Directed Graphs and Polynomial Representations
G 1
2
1
b
a
463
111
111
112
112
121
121
211
bac abc
122
2
c
211
bca
212
122 212
acb 221
221
cab
222
222
cba
3
Figure 2: Digraph G and corresponding digraph X⊗ [G]. Generating functions preserve the operations sum, product, and tensor product (R + S)(X) = R(X) + S(X),
(3.5)
(R · S)(X) = R(X) · S(X),
(3.6)
(R ⊗ S)(X) = R(X) ⊗ S(X).
(3.7) (w)
Example 3.4. Let X be the singleton matix species X : Gn → Gn ,→ Gn ( G, if |Arc(G)| = 1, X[G] = 0, otherwise.
(3.8)
(ω)
k
The tensor power X⊗ : Gn → G[n]k ,→ G[n]k is given by the formula k
X⊗ [G] =
∑
X[G1 ] ⊗ · · · ⊗ X[Gk ]
∑
G1 ⊗ · · · ⊗ G k .
(3.9)
G1 +···+Gk =G |Arc(G1 )|=···=|Arc(Gk )|=1
=
(3.10)
G1 +···+Gk =G |Arc(G1 )|=···=|Arc(Gk )|=1 k
More explicitly, the arcs of X⊗ [G] are the total orders of length k on Arc(G), and k l = (a1 , a1 , . . . , ak ) ∈ Arc(X⊗ [G]) goes from vertex i to vertex j, i, j ∈ [n]k , iff ar goes k from ir to jr in G, r = 1, 2, . . . , k (see Figure 2). Observe that X⊗ [G] is the empty digraph if |Arc(G)| 6= k.
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4. Eulerian Bijections and Schur’s Product Rule For a digraph G with vertices in [n] the outdegree and indegree sequences are respectively defined by od(G) = α = (α1 , α2 , . . . , αn ), where αi = |tG−1 (i)|, i ∈ [n], id(G) = β = (β1 , β2 , . . . , βn ), where βi = |h−1 G ( j)|, j ∈ [n]. G is said to be balanced if α = β. Definition 4.1. Let G1 and G2 be two digraphs in Gn . A bijection κ : Arc(G1 ) → Arc(G2 ) is called Eulerian (from G1 to G2 ) if it satisfies tG2 (κ(a)) = hG1 (a) for every a in Arc(G1 ). An Eulerian permutation is an Eulerian bijection from a digraph to itself. We denote by Eul[G1 , G2 ] the set of Eulerian bijections from G1 to G2 and by S [G] the set of Eulerian permutations on G. For a composition α = (α1 , α2 , . . . , αn ), denote by α! the product α1 !α2 ! . . . αn !. Obviously, ( β!, if id(G1 ) = od(G2 ), with β = id(G1 ), |Eul[G1 , G2 ]| = 0, otherwise. In particular |S [G]| =
(
α!,
if G is balanced with od(G) = id(G) = α,
0,
otherwise.
Definition 4.2. Let G1 and G2 two digraphs as above, and κ ∈ Eul[G1 , G2 ]. The composition G2 ◦κ G1 is the digraph whose arcs are the paths of length two of the form (a, κ(a)), a ∈ G1 . Formally, if we denote by G3 the composition digraph G2 ◦κ G1 , we have Arc(G3 ) = {(a, κ(a))|a ∈ Arc(G1 )}, tG3 (a, κ(a)) = tG1 (a), and hG3 (a, κ(a)) = hG2 (κ(a)), for every a ∈ Arc(G2 ). (ω)
Theorem 4.3. Let R : Gn → GV be a matrix species. R(X) satisfy equation (3.2) iff for every pair of digraph G1 and G2 in Gn we have that the weighted digraphs R[G1 ] ? R[G2 ] and ∑κ∈Eul[G1 , G2 ] R[G2 ◦κ G1 ] have the same weight matrix. Remark 4.4. The sum over Eul[G1 , G2 ] is assumed to be the empty digraph in the / category GV when Eul[G1 , G2 ] = 0. The proof of this theorem is simple by making use of the operation of substitution of multisort matrix species. It is done in the appendix of this article. As a corollary we get a very simple combinatorial rule to multiply two coefficient matrices of the generating function of an invariant matrix species (see Figure 3). Corollary 4.5. Let R be a matrix species. R(X) satisfy equation (3.2) iff for every pair of digraphs G1 , G2 ∈ Gn we have R G1 · R G2 =
∑
κ∈Eul[G1 , G2 ]
R G2 ◦ κ G1 .
(4.1)
Directed Graphs and Polynomial Representations
R
1
a b
2
c
1
. R
2
1
2
d 1
e
= R
+R
(a,d)
1
2
f
465
(b,e)
2
1
(c,f)
2
(a,d) 1 1 (c,e) (b,f) 2 2
Figure 3: Product rule for invariant matrix species. We easily obtain the following corollary. Corollary 4.6. Let R be as above. If od(G2 ) 6= id(G1 ) then RG1 ·RG2 = 0. In particular, if G is not balanced, R2G = 0. In terms of adjacency matrices we have RA · RB = 0 if the row sum sequence of B (row(B)) is different from the column sum sequence of A (col(A)). If row(A) 6= col(A) then R2A = 0. Let A, B and C be three integer matrices. Introducing the coefficient C = |{κ ∈ Eul[G1 , G2 ]|A (G2 ◦κ G1 ) = C}|, A, B where G1 and G2 are fixed digraphs satisfying A (G1 ) = A and A (G2 ) = B, we can rewrite equation (4.1) as follows C R . (4.2) RA · RB = ∑ A, B C C 5. The Digraph Version of the Schur Algebra The product rule motivates the following definition. (k)
Definition 5.1. Let Gn be the category of digraphs which have exactly k arcs. For (k) e its isomorphism class (that can be identified with its a digraph G in Gn denote by G (k) (k) adjacency matrix A (G)). Denote by Gn the set of the isomorphism classes in Gn . (k) The digraph algebra DC (n, k) is the vector space C · Gn endowed with the product e1 · G e2 = G
∑
κ∈Eul[G1 , G2 ]
G^ 2 ◦κ G1 .
(5.1)
Green re-introduced in [5] the Schur algebra SF (n, k) over an infinite field F as the dual of the coalgebra Fk [X] of homogeneous of degree k polynomials in a matrix of variables X, with coproduct ∆(p(X)) = p(X · Y) ∈ Fk [XY] ,→ Fk [X] ⊗ Fk [X]. and counit
ε(p(X)) = p(In ),
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M.A. M´endez
where In is the n × n identity matrix. See also the discussion in [7, pp. 5–6] about the historical motivations for the introduction of Schur algebras. For i, j ∈ [n]r , he defined the symbols ξi, j as the dual basis of the monomials r
xi, j = ∏ xik , jk . k=1
In our notation xi, j = XA , where ah, l is the number of k0 s such that ik = h and jk = l, and ξA is defined by ( 1, if A = A0 , A0 ξA (X ) = 0, otherwise. Proposition 5.2. DC (n, k) is isomorphic to the Schur algebra SC (n, k) via the map e→ φ: G 7 A!ξA ,
where A = A (G). Proof. Let
C! C C := . A, B A!B! A, B
Let G1 and G2 be digraphs such that A (G1 ) = A and A (G2 ) = B. We have that C f1 · G f2 ) = ^ φ(G φ( G ◦ G ) = 2 κ 1 ∑ ∑ A, B C!ξC , C κ∈Eul[G , G ] 1
2
f1 ) · φ(G f2 ) = A! · B!ξA · ξB . φ(G
By duality and Proposition 7.9 in appendix, we obtain that C ξ , ξA · ξB = ∑ A, B C C
(5.2)
f1 · G f2 ) = φ(G f1 ) · φ(G f2 ). which means that φ(G
Equations (5.2) and (7.5) give a very simple combinatorial rule for the product of two basic elements of the Schur algebra. To multiply ξA · ξB we: • Identify ξA and ξB respectively with ξG1 and ξG2 , G1 and G2 being digraphs with respective adjacency matrices A and B, but making parallel arcs undistinguishable (see Figure 4). • Compute all the composition digraphs G3 whose arcs are the arc-disjoint paths of length two, first arc in G1 , second arc in G2 . • The coefficient of ξG3 in the expansion of ξG1 · ξG2 is the number of matrices of words of the form (wi, j )ni, j=1 , where wi, j is a word over the multiset of i-to- j arcs in G3 .
Directed Graphs and Polynomial Representations
·ξ
ξ
a c
467
= 3ξ
a
d d f
b
+ 2ξ
(a,d) (a,d) (b,f)
e
(a,d) (b,f) (c,d) (a,e)
(c,e)
Figure 4: Product rule for Schur algebras.
For example, from Figure 4 we get ξ21112, 1121 = ξ2
21 10
!
= 3ξ 3 0! + 2ξ 2 1! = 3ξ1112, 1112 + 2ξ1112, 1121. 01 10
The coefficients 3 and 2 appear because the number of words that can be formed with the multisets of symbols {(a, d), (a, d), (b, f )} and {(a, d), (b, f )} are respectively 3 and 2. Green has given a more algebraic calculating tool in [4]. Definition 5.3. Let G be a digraph. We say that G is a functional digraph if t G is injective. The partial function f : [n] → [n], f = hG ◦ tG−1 , characterizes G up to isomorphism. We denote G as G f . The composition of f , Cp( f ), is defined as the tuple of sets ( f −1 (1), f −1 (2), . . . , f −1 (n)). If in addition hG is also injective we say that G f a bijective digraph. We denote by C f , g be the set of (partial) bijections σ satisfying σ(Cp( f )) = (σ( f −1 (1)), σ( f −1 (2)), . . . , σ( f −1 (n))) = Cp(g), or equivalently f ◦ σ = g. Definition 5.4. The transpose Gt of a digraph G is obtained by reversing the arrows of G. Formally, Arc(Gt ) = Arc(G), tGt = hG , hGt = tG . Definition 5.5. A loop digraph G is one satisfying tG = hG . Every loop digraph is balanced. A loop digraph with outdegree α is unique up to isomorphism, we shall denote it by Lα (see Figure 5). We obtain the following identities in DC (n, k) by simple inspection. Proposition 5.6. Let f , g : I → [n] be functions from a subset I of [n] with |I| = k, and such that id(G f ) = od(Gtg ). We have ft · G ff = L fα , G f
(5.3)
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M.A. M´endez
(a)
(b)
(c)
Gt f
Gf
L (2,0,1,0)
Figure 5: (a) Functional digraph, (b) its transpose, (c) loop digraph. where α = id(G f ) (see Figure 5), e= fα · G L e·L fα = G
In particular
ft = ff · G G g
( (
e α!G,
if od(G) = α,
0,
otherwise,
e α!G,
if id(G) = α,
0,
∑
σ∈C f , g
ft = ff · G G f
(5.4)
(5.5)
otherwise,
fσ . G
∑
σ∈fix(Cp( f ))
(5.6)
fσ , G
(5.7)
where fix(Cp( f )) = C f , f is the Young group of permutations fixing each block of Cp( f ). For a bijective digraph Gσ such that id(Gσ ) = od(G f ) we have fσ · G ff = G ] G f ◦σ ,
t ft · G fσ = G^ G . f f ◦σ−1
(5.8) (5.9) (k)
Lemma 5.7 (Factorization Lemma). Let G be a digraph in Gn , 1 ≤ k ≤ n. Then, there exist functions f , g : [k] → [k] such that ft · G e=G fg . G f
(5.10)
ft · G e=G fσ · G ff . G f
(5.11)
In particular, if G is balanced, there exist a permutation σ ∈ Sk such that
Directed Graphs and Polynomial Representations
1 2
1 1 2 . 2
1 2
3
3
3
~ G ft
3
.
~
469
=
=
Gf
1 2
1 2
3
3
~ L (2,1,0)
ft · G ff . Figure 6: The product G f
1 2
1 2
1 . 2
1 2
3
3
3
3
=
1 2 3
1 1 2 + 2 3 3
1 2 3
ft . ff · G Figure 7: The product G g
Proof. Let {a1 , a2 , . . . , ak } be a total order of the elements of Arc(G). Let f and g defined as follows, f (i) = tG (ai ), g(i) = hG (ai ) (1 ≤ i ≤ k). ft and G fg defined by Let Gtf and Gg be respectively the representatives of G f Arc(Gtf ) = Arc(Gg ) = Arc(G),
tGtf = tG , hGtf (ai ) = i = tGg (ai )(1 ≤ i ≤ k), hGg = hG . Since the only element in Eul[Gtf , Gg ] is the identity ι : [k] → [k], we have ft G ^t f G f g = Gg ◦ι G f .
Then, all we have to do is to verify that G1 = Gg ◦ι Gtf is isomorphic to G. It is trivial to check that ψ : Arc(G1 ) → Arc(G), ψ(ai , ai ) = ai (1 ≤ i ≤ k) is a digraph isomorphism. If G is balanced, od(Gtf ) = id(Gg ), i.e., (| f −1 (1)|, | f −1 (2)|, . . . , | f (−1) (n)|) = (|g−1 (1)|, |g−1 (2)|, . . . , |g(−1) (n)|). Then, there exists a permutation σ ∈ Sk such that σ(g−1 (i)) = f −1 (i) (1 ≤ i ≤ k), i.e., fg = G fσ · G ff . g = f ◦ σ. By equation (5.8), we have G
470
M.A. M´endez
~ a1
1 1 2= 2 3 3
1 a 2 2 a3 3 1 = 2 3
~ a1 a
2
a
3
~ a 1
a
2
a
3
1 1 2 . 2 3
a2 a3
3
1 1 2. 2
1 1 2. 2
3
3 3
3
~ a1 ~ a1 a2 a3
1 2 3 1 2 3
ft · G e=G fσ · G ff . Figure 8: Factorization G f
Remark 5.8. Note that any permutation σ satisfying equation (5.11) defines in a natural ˆ i ) = aσ(i) , way an Eulerian permutation σˆ : Arc(G) → Arc(G), σ(a ˆ i )) = tG (aσ(i) ) = f (σ(i)) = ( f ◦ σ)(i) = g(i) = hG (ai ). tG (σ(a ft · Gg e=G f Moreover, for any permutation τ ∈ fix(Cp( f )) we have G τ◦σ · G f . Since f |fix(Cp( f ))| = od(G)! = |S [G]| we obtain
S [G] = {τd ◦ σ|τ ∈ fix(Cp( f ))}.
(5.12)
Corollary 5.9. Let R be an invariant matrix species. Then, the trace of the matrix R G , trRG can be computed by using the following recipe ( if G is balanced, ∑σ∈ ˆ S [G] trRGσ , (5.13) trRG = 0, otherwise. Proof. If G is not balanced, by Corollary 4.6 R2G = 0, and hence trRG = 0. If G is balanced, by equations (5.11) and (5.7) we obtain trRG = trRGtf · RGσ · RG f = trRGσ · RG f · RGtf =
∑
τ∈fix(Cp( f ))
trRGσ · RGτ =
The result follows from the previous remark.
∑
τ∈fix(Cp( f ))
trRGτ◦σ .
Directed Graphs and Polynomial Representations
471
{1,2}
{1,2} 1 2
1 2
3
3
G
-1
{1,3}
{1,3}
{2,3}
{2,3}
Figure 9: Action of E(2) over G. 6. Examples of Invariant Matrix Species In this section we give combinatorial models for the polynomial matrices representing the symmetric and exterior powers. Using Corollary 4.5 we give purely combinatorial proofs of the invariance of these matrices. 6.1. The Exterior Power Let Pk [n] denote the class of k-subsets of [n]. Consider the matrix species (ω)
E(k) : Gn → GP [n] k
assigning to a digraph G the weighted digraph E(k) [G] defined as follows; if G is not a (ω) bijective digraph or |Arc(G)| 6= k, E(k) [G] is the empty digraph in GP [n] . Otherwise, G k is equal to a bijective digraph of the form Gσ for some bijection σ : I → J, (I, J ∈ Pk [n]). In such case E(k) [Gσ ] = (G0 , w), where Arc(G0 ) = {σ}, tG (σ) = I, hG (σ) = J, and w(σ) = sign(σ). It is easy to check that the generating function of E(k) (X) is the k-compound matrix X(k) = (detI, J (X))I, J∈Pk [n] . The invariance of E(k) is the Binnet-Cauchy theorem. We give a combinatorial proof of it. Proposition 6.1. The matrix species E(k) is invariant. Proof. E(k) (In ) is clearly equal to IPk [n] . By Theorem 4.3 we only have to prove that for every pair G1 , G2 ∈ Gn |E(k) [G1 ] ? E(k) [G2 ]| = |
∑
κ∈Eul[G1 , G2 ]
E(k) [G2 ◦κ G1 ]|.
(6.1)
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M.A. M´endez
The product E(k) [G1 ] ? E(k) [G2 ] is different from the empty digraph iff G1 = Gσ and G2 = Gτ are both bijective and the image J ⊆ [n] of σ is equal to the domain of τ. Let I and K be respectively the domain of σ and the image of τ. E(k) [Gσ ] ? E(k) [Gτ ] has only one arc (σ, τ) which goes from I to K and whose weight is sign(σ) · sign(τ). The digraph ∑κ∈Eul[Gσ , Gτ ] E(k) [Gτ ◦κ Gσ ] = E(k) [Gτ◦σ ] has also only one arc τ ◦ σ from I to J, and weight sign(τ ◦ σ) = sign(τ) · sign(σ). Obviously |E(k) [Gτ◦σ ]| = |E(k) [Gσ ] ? E(k) [Gτ ]|. To finish the proof we have to check that ∑κ∈Eul[G1 , G2 ] E(k) [G2 ◦κ G1 ] is zero for all other kinds of digraphs G1 and G2 . Since E(k) [G2 ◦κ G1 ] is the empty digraph if G2 ◦κ G1 is not bijective, the only relevant case we have to consider is when G 1 = G f is a non-bijective functional digraph with k arcs, and G2 is of the form G2 = Gtg with id(G f ) = od(Gtg ). Let Gˆ w = ∑ E(k) [G f ◦κ Gtg] = ∑ E(k) [Gσ ], σ∈C ( f , g)
κ∈Eul[G f , Gtg ]
ˆ = C ( f , g), all the arcs of Gˆ go the last identity obtained by equation (5.6). Then Arc(G) ˆ from domain of f to domain of g, and the weight of σ is sign(σ), for every σ in Arc( G). Let i, j be two elements of [n] such that f (i) = f ( j). The function ψi, j (σ) = σ ◦ (i, j) is a sign reversing involution on the set of arcs C ( f , g). Then |Gˆ w | = 0. 6.2. The Symmetric Power Definition 6.2. A flow over a digraph G is an n-uple (l1 , l2 , . . . , ln ), where each li is a total order on the (possibly empty) set tG−1 (i) of arcs leaving i, 1 ≤ i ≤ n. Denote by Mk [n] the set of n-compositions of k. An element α = (α1 , α2 , . . . , αn ) ∈ Mk [n] can be identified with a multiset h1α1 , 2α2 , . . . , nαn i of total cardinality k. Consider the matrix species (ω) S(k) : Gn → GMk [n] ,→ GM [n] k
assigning to a digraph G the digraph S(k) [G] defined as follows; if |Arc(G)| 6= k, S(k) [G] is the empty digraph. Otherwise S(k) [G] is the digraph whose arcs are the flows of G, (k) all the arcs going from α = od(G) to β = id(G). The generating function of S α, β (X) is (k) Sα, β (X)
n
= [t ] ∏ β
i=1
n
∑ xi, rtr
r=1
!αi
,
(see [8, exercise 3.1]). Then, S(k) (X) is the matrix corresponding to the symmetric power (see [3]). Proposition 6.3. The matrix species S(k) is invariant. Proof. As in the previous proposition, we have to prove that |S(k) [G1 ] ? S(k) [G2 ]| = |
∑
κ∈Eul[G1 , G2 ]
S(k) [G2 ◦κ G1 ]|.
(6.2)
Directed Graphs and Polynomial Representations
473
(3,0) 1 2
a b c
(3,0) (ab,c)
1
(2,1)
2
(1,2)
G
(2,1) (ba,c)
(0,3)
(1,2) (0,3)
Figure 10: Action of S(3) over G. The left and the right hand side are both zero when id(G1 ) 6= od(G2 ). Assume that od(G1 ) = α, id(G1 ) = od(G2 ) = β, and id(G2 ) = γ. There are α! arcs from α to β in S(k) [G1 ] and β! arcs from β to γ in S(k) [G2 ]. By the definition of star product, there are α! × β! arcs from α to γ in |S(k) [G1 ] ? S(k) [G2 ]|. By the definition of composition of digraphs, od(G2 ◦κ G1 ) = α, and id(G2 ◦κ G1 ) = γ for every Eulerian bijection κ. Then, there are α! arcs from α to γ in S(k) [G2 ◦κ G1 ], for every κ ∈ Eul[G1 , G2 ]. Since |Eul[G1 , G2 ]| = β! we obtain (6.2). 7. Appendix 7.1. Two-Sort Matrix Species Definition 7.1. A two-sort matrix species is a functor M fron the product category Gn1 × Gn2 to Gn . Analogously to the ordinary case, the generating function of M is defined as XA YB |M[A, B]| M(X, Y) = . ∑ A!B! A∈Nn1 ×n1 , B∈Nn2 ×n2 Definition 7.2. Let M1 and M2 be two two-sort matrix species M1 , M2 : G n1 × G n2 → G n . We define the product M1 · M2 by the recipe M1 · M2 [G1 , G2 ] =
∑
G01 +G001 =G1 G02 +G002 =G2
M1 [G01 , G02 ] ? M2 [G001 , G002 ].
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M.A. M´endez
a
1
1
1
d
1
1 (a,d)
2 2
c
2
(c,f)
