2.1 Tracts

A tract $F = (G, N_F)$ is an abelian group G (written multiplicatively), together with an additive relation structure $N_F$ , which is a subset of the group semiring $\mathbb {N}[G]$ satisfying:

1. (T1) The zero element $0$ of $\mathbb {N}[G]$ belongs to $N_F$ .

2. (T2) The identity element $1$ of G does not belong to $N_F$ .

3. (T3) There is a unique element $\epsilon $ of G such that $1 + \epsilon \in N_F$ .

4. (T4) If $g \in G$ and $a \in N_F$ , then $ga \in N_F$ .

We think of $N_F$ as linear combinations of elements of G which ‘sum to zero’ and call it the null set of the tract F.

A useful lemma from [Reference Baker and Bowler2] about tracts is as follows.

Because of this lemma, we also write F for the set $G \cup \{0\}$ , and write $-1$ instead of $\epsilon $ . We will sometimes use G and $F^\times $ interchangeably.

A tract homomorphism $\varphi : F_1 \to F_2$ is a group homomorphism $\varphi : F_1^\times \to F_2^\times $ such that the induced semiring homomorphism $\mathbb {N}[F_1^\times ] \to \mathbb {N}[F_2^\times ]$ maps $N_{F_1}$ to $N_{F_2}$ . All tracts together with tract homomorphisms between them form a category. An involution $\tau $ of a tract F is a tract homomorphism $\tau : F \to F$ such that $\tau ^2$ is the identity map.

We list some examples of tracts and tract homomorphisms in Figure 1.

Figure 1

Examples of tracts and tract homomorphisms.

The category of tracts admits products, which will be useful for studying representations of matroids and orthogonal matroids in Section 5. Let $F_1, F_2$ be tracts. The (categorical) product $F_1 \times F_2$ can be constructed explicitly as follows. As a set, $F_1 \times F_2$ is $(F_1^\times \oplus F_2^\times ) \cup \{0\}$ , endowed with the coordinate-wise multiplication on $F_1^\times \oplus F_2^\times $ , and the rule $0 \cdot (x_1, x_2) = (x_1, x_2) \cdot 0 = 0$ . The null set of $F_1 \times F_2$ is

$$\begin{align*}N_{F_1 \times F_2} := \left\{ \sum_{i=1}^k (x_i,y_i) \in {\mathbb N}[F_1^\times \oplus F_2^\times] : \sum_{i = 1}^k x_i \in N_{F_1} \text{ and } \sum_{i = 1}^k y_k \in N_{F_2}\right\}. \end{align*}$$

2.2 Grassmann–Plücker functions

The easiest way of defining matroids over tracts is via the Grassmann–Plücker functions. This is also the tract analogue of the basis exchange axiom for ordinary matroids.

Two strong (resp. weak) Grassmann–Plücker functions $\varphi _1$ and $\varphi _2$ are equivalent if $\varphi _1 = c \cdot \varphi _2$ for some $c \in F^\times $ , and we call an equivalence class $M_\varphi := [\varphi ]$ of strong (resp. weak) Grassmann–Plücker functions a strong (resp. weak) matroid over the tract F, or simply a strong (resp. weak) F-matroid. It can be shown that every strong F-matroid is a weak F-matroid. We denote by $\underline {M}_\varphi $ the underling ordinary matroid of a strong or weak F-matroid $M_\varphi $ whose set of bases is $\mathcal {B}(\underline {M}_\varphi ) = \{\{x_1,\dots ,x_r\} \in \binom {E}{r}: \varphi (x_1,\dots ,x_r) \ne 0\}$ .

2.3 F-circuits and dual pairs

We now give two cryptomorphic definitions of matroids over a tract F in terms of F-circuits and dual pairs of F-signatures.

Denote by $F^E$ the set of all functions from E to F. The support of $X \in F^E$ is the set of elements e in E such that $X(e) \ne 0$ , and is denoted by $\underline {X}$ or $\mathrm {Supp}{X}$ . Given two functions $X = (x_1, \dots , x_n)$ and $Y = (y_1, \dots , y_n) \in F^E$ , where F is endowed with an involution $x \mapsto \overline {x}$ , the inner product of X and Y is $X \cdot Y := \sum _{k = 1}^n x_k \overline {y_k}$ . We say that two functions X and Y are orthogonal, denoted by $X \perp Y$ , if $X \cdot Y \in N_F$ .

When F is the field $\mathbb {C}$ of complex numbers or the sixth-root-of-unity partial field $R_6$ , the involution $x \mapsto \overline {x}$ should be taken to be the complex conjugation. For $F \in \{\mathbb {K}, \mathbb {S}, \mathbb {T}\}$ , the involution should be taken to be the identity map.

The linear span of $X_1, \dots , X_k \in F^E$ is defined to be the set of all functions $X \in F^E$ such that

$$\begin{align*}c_1X_1 + c_2X_2 + \dots + c_kX_k - X \in (N_F)^E \end{align*}$$

for some $c_1,\dots ,c_k \in F$ .

For an ordinary matroid $\underline {M}$ on E, we denote by $C_{\underline {M}}(B,e)$ the fundamental circuit of M with respect to $B\in {\mathcal B}(\underline {M})$ and $e\in E\setminus B$ . The subscript will be omitted if no confusion arises.

2.4 F-vectors

One can naturally ask for an axiomatization of linear spaces over a tract F. Anderson answered this question and gave another cryptomorphic definition of strong F-matroids in terms of F-vectors in [Reference Anderson1].

For a subset $\mathcal {W} \subseteq F^E$ , a support basis for $\mathcal {W}$ is a minimal subset of E meeting every element of $\mathrm {Supp}(\mathcal {W} \setminus \{0\})$ . A reduced row-echelon form of $\mathcal {W}$ with respect to a support basis B is a subset $\Phi _B = \{w_i^B\}_{i \in B} \subseteq \mathcal {W}$ such that $w_i^B(j) = \delta _{ij}$ for each $i,j\in B$ , and every $w \in {\mathcal W}$ is in the linear span of $\Phi _B$ . It is not difficult to see that if $\Phi _B$ exists, then it is unique. We say a collection $\Phi = \{\Phi _B\}$ of reduced row-echelon forms is tight if $\mathcal {W}$ is precisely the set of elements of $F^E$ which are in the linear span of $\Phi _B$ for all $\Phi _B \in \Phi $ .

2.5 Cryptomorphisms

The main results of [Reference Baker and Bowler2, Reference Anderson1] are the following theorems.

2.6 Functoriality, duality and minors

Let F be a tract with an involution $x \mapsto \overline {x}$ . The theory of functoriality, duality and minors for matroids over tracts generalizes the corresponding classical theory for matroids. For simplicity, here we only give the descriptions via the strong Grassmann–Plücker functions.

Given a strong Grassmann–Plücker function $\varphi : E^r \to F$ and a tract homomorphism $f: F \to F'$ , we define the pushforword $f_\ast \varphi : E^r \to F'$ as

$$\begin{align*}(f_\ast\varphi)(x_1, \dots, x_r) = f(\varphi(x_1, \dots, x_r)). \end{align*}$$

It is not hard to see that $f_\ast \varphi $ is a strong Grassmann–Plücker function. Notice that pushforwards are functorial: if $F_1 \xrightarrow {f} F_2 \xrightarrow {g} F_3$ are tract homomorphisms, then $(g \circ f)_\ast = g_\ast \circ f_\ast $ .

The dual Grassmann–Plücker function $\varphi ^\ast : E^{n-r} \to F$ of $\varphi $ is determined by (GP2) and

$$\begin{align*}\varphi^\ast(x_1, \dots, x_{n-r}) = \mathrm{sgn}(x_1, \dots, x_{n-r}, x_1', \dots, x_r') \cdot \overline{\varphi(x_1', \dots, x_r')}, \end{align*}$$

where $x_1', \dots , x_r'$ is any ordering of $E \setminus \{x_1, \dots , x_{n-r}\}$ , and $\mathrm {sgn}(x_1, \dots , x_{n-r}, x_1', \dots , x_r') \in \{\pm 1\}$ is the permutation sign taken as an element of F. This notion of dual Grassmann–Plücker functions satisfies $\varphi ^{\ast \ast } = \varphi $ , and the underlying matroid of $\varphi ^\ast $ is the dual matroid of the underlying matroid of $\varphi $ .

Let $\varphi : E^r \to F$ be a strong Grassmann–Plücker function with the underlying matroid $\underline {M}_{\varphi }$ and let $A \subseteq E$ . We denote by $\ell $ and k the ranks of A and $E\setminus A$ in $\underline {M}_\varphi $ , respectively.

Let $\{a_1, \dots , a_\ell \}$ be a maximal independent subset of A in $\underline {M}_{\varphi }$ . The contraction $\varphi / A: (E \setminus A)^{r - \ell } \to F$ is defined by

$$\begin{align*}(\varphi / A)(x_1, \dots, x_{r-\ell}) = \varphi(x_1, \dots, x_{r - \ell}, a_1, \dots, a_r). \end{align*}$$

Choose $\{b_1, \dots , b_{r-k}\}$ such that $\{b_1, \dots , b_{r-k}\}$ is a basis of $\underline {M}_{\varphi } / (E\setminus A)$ . Then the deletion $\varphi \setminus A: (E \setminus A)^k \to F$ is defined by

$$\begin{align*}(\varphi \setminus A)(x_1, \dots, x_k) = \varphi(x_1, \dots, x_k, b_1, \dots, b_{r-k}). \end{align*}$$

The following lemma shows that the contractions and deletions are well-defined.

3.1 Wick functions

We describe the first cryptomorphic characterization of strong, moderately weak and weak orthogonal matroids over tracts in terms of Wick functions. We denote by $\mathcal {T}_n$ the family of all transversals of E.

Two strong Wick functions $\varphi $ and $\psi $ with coefficients in F are equivalent if $\varphi = c \psi $ for some nonzero $c \in F$ , and we call an equivalence class $M_\varphi = [\varphi ]$ of strong Wick functions a strong orthogonal matroid over the tract F, or simply a strong orthogonal F-matroid. We similarly define (moderately) weak orthogonal F-matroid. It is trivial that every moderately weak orthogonal F-matroid is weak. Proposition 3.2 shows that every strong orthogonal F-matroid is a moderately weak orthogonal F-matroid. The three notions of orthogonal F-matroids are the same when F is a partial field [Reference Baker and Jin3], the tropical hyperfield ${\mathbb T}$ [Reference Rincón23] or the Krasner hyperfield ${\mathbb K}$ . We denote by $\underline {M}_\varphi $ the underlying orthogonal matroid of the orthogonal F-matroid $M_\varphi $ whose set of bases is $\mathrm {Supp}(\varphi )$ .

Proof. Let $\varphi : [n]^r \to F$ be a strong Grassmann–Plücker function. Define $\psi : {\mathcal T}_n \to F$ to be $\psi (T) := \varphi (a_1, \dots , a_r)$ if $T = B \cup ([n] \setminus B)^\ast $ for $B = \{a_1 < \cdots < a_r\}$ , and $\psi (T) = 0$ otherwise. It is obvious that $\psi $ is not identically zero, and we claim that $\psi $ satisfies (W2). To prove (W2), we take $T_1, T_2 \in {\mathcal T}_n$ with $(T_1 \triangle T_2) \cap [n] = \{x_1 < \cdots < x_m\}$ . Suppose without loss of generality that $T_1 \cap [n] = \{b_1 < \dots < b_{r+1}\}$ and $T_2 \cap [n] = \{c_1 < \dots < c_{r-1}\}$ . If $x_k \in (T_2 \setminus T_1) \cap [n]$ , then $\psi (T_1 \triangle \{ x_k , x_k ^* \}) = \psi (T_2 \triangle \{ x_k , x_k ^* \}) = 0$ . If $x_k = b_j \in (T_1 \setminus T_2) \cap [n]$ , then since $|T_1 \cap [x_k]| = j$ and $|T_2 \cap [x_k]| = k - j + 2|T_1 \cap T_2 \cap [x_k]|$ , we have

$$ \begin{align*} \psi(T_1 \triangle \{ x_k , x_k ^* \}) = \varphi(b_1, \dots, \hat{b_j}, \dots, b_{r+1}) \,\text{ and }\, \psi(T_2 \triangle \{ x_k , x_k ^* \}) = (-1)^{k-j} \varphi(b_j, c_1, \dots, c_{r-1}). \end{align*} $$

It follows that

$$ \begin{align*} \sum_{k = 1}^m (-1)^k \psi(T_1 \triangle \{ x_k , x_k ^* \}) \psi(T_2 \triangle \{ x_k , x_k ^* \}) = \!\sum_{j = 1}^{r+1} (-1)^{j} \varphi(b_1, \dots, \hat{b_j}, \dots, b_{r+1})\varphi(b_j, c_1, \dots, c_{r-1}) \in N_F. \end{align*} $$

Therefore, $\psi $ is a strong Wick function whose support forms the bases of $\mathrm {lift}(\underline {M}_\varphi )$ .

Conversely, let $\psi $ be a strong Wick function on $E = [n] \cup [n]^\ast $ such that all elements of $\{B \cap [n] : B \in \mathrm {Supp}(\psi )\}$ have the same cardinality r. Let $\varphi : [n]^r \to F$ be the (unique) function satisfying (GP1) and (GP2) defined by $\varphi (a_1, \dots , a_r) := \psi (T)$ , where $T = \{a_1, \dots , a_r\} \cup ([n] \setminus \{a_1, \dots , a_r\})^\ast $ for all $\{a_1 < \cdots < a_r\} \subseteq [n]$ . Take $J_1 = \{b_1 < \cdots < b_{r+1}\}, J_2 = \{c_1, \dots , c_{r-1}\} \subseteq [n]$ , and write $J_1' = J_1 \cup ([n] \setminus J_1)^\ast $ and $J_2' = J_2 \cup ([n] \setminus J_2)^\ast $ . Then

$$ \begin{align*} \sum_{j = 1}^{r+1} (-1)^j \varphi(b_1, \dots, \hat{b_j}, \dots, b_{r+1}) \varphi(b_j, c_1, \dots, c_{r-1}) = \sum_{j = 1}^{r+1} (-1)^j \cdot \psi(J_1' \triangle \{ b_j , b_j ^* \}) \cdot (-1)^{m_j}\psi(J_2' \triangle \{ b_j , b_j ^* \}), \end{align*} $$

where $m_j$ is the number of elements in $J_2$ that are less than $b_j$ . Write $(J_1' \triangle J_2') \cap [n] = \{x_1, \dots , x_m\}$ . If $e \in J_2$ , then since all elements of $\{B \cap [n] : B \in \mathrm {Supp}(\psi )\}$ have cardinality r, we have $\psi (J_2' \triangle \{ e , e ^* \}) = 0$ . Therefore, we have

$$ \begin{align*} \sum_{j = 1}^{r+1} (-1)^j \varphi(b_1, \dots, \hat{b_j}, \dots, b_{r+1}) \varphi(b_j, c_1, \dots, c_{r-1}) = \!\sum_{k = 1}^m (-1)^k \psi(J_1' \triangle \{ x_k , x_k ^* \})\psi(J_2' \triangle \{ x_k , x_k ^* \}) \in N_F. \end{align*} $$

It’s not hard to see that the two constructions are inverses of each other.

3.2 Orthogonal signatures and circuit sets

Let $\underline {M}$ be an ordinary orthogonal matroid on E. As in Section 2, we denote the support of $X \in F^E$ by $\underline {X} = \{i \in E: X(i) \neq 0\}$ . If $X \in F^E$ , we write $X^\ast \in F^E$ for the function defined by $X^*(i) := X(i^*)$ . Notice that this induces an obvious involution $\ast $ on the subsets of $F^E$ .

The inner product $\langle \cdot , \cdot \rangle $ on $F^E$ with respect to the involution $x \mapsto \overline {x}$ is defined to be

$$ \begin{align*} \langle X, Y \rangle = \sum_{i\in [n]} ( X(i) \overline{Y(i)} + \overline{X(i^\ast)} Y(i^\ast)). \end{align*} $$

Note that $\langle Y, X \rangle = \overline {\langle X, Y \rangle }$ . Let $\tilde {\cdot }: F^E \to F^E$ be such that $\tilde {X}(i) = X(i)$ if $i\in [n]$ and $\tilde {X}(i) = \overline {X(i)}$ otherwise. Then $\langle X, Y^\ast \rangle = \sum _{i \in E} \tilde {X}(i) \tilde {Y}(i^\ast )$ .

We say that an F-signature ${\mathcal C}$ of $\underline {M}$ satisfies the $2$ -term orthogonality if the following holds:

1. (O₂) $\langle X, Y^* \rangle \in N_F$ for all $X,Y \in {\mathcal C}$ with $|\underline {X} \cap \underline {Y}^*| = 2$ ,

Notice that both (L-i) $'$ and (L-ii) $'$ follow from (L). To show that (L) implies (L-i) $'$ , we consider $f \in \underline {X_1} \cap \underline {X_2}$ as given in (L-i) $'$ . By (C3), there exists a circuit $C \subseteq (\underline {X_1} \cup \underline {X_2}) \setminus \{f\}$ . Observe that $e_1,e_2 \in C$ ; otherwise, C is a proper subset of $\underline {X_2}$ or $\underline {X_1}$ , a contradiction. Let Y be the F-circuit whose support is C. Applying (L), we see that $\tilde {Y}$ is in the linear span of $\{\tilde {X_e}\}_{e\in B^*}$ . We claim that the coefficients for terms other than $\tilde {X_1}$ and $\tilde {X_2}$ must be zero. In fact, if $e \in B^\ast $ but $e \ne e_1, e_2$ , then $e \in C(B, e) \setminus C$ . As a result, the coefficient for $\tilde {X_e}$ is zero. Therefore, $\tilde {Y}$ is in the linear span of $\tilde {X_1}$ and $\tilde {X_2}$ .

For (L-ii) $'$ , we consider $\underline {X_i} = C(B, e_i)$ for $i = 1,2,3$ as given in (L-ii) $'$ . Since $\underline {X_1} \cup \underline {X_2}$ is not admissible, $X_1(e_2^*) \ne 0$ and $X_2(e_1^*) \ne 0$ . Then $B' := B\triangle \{e_1,e_1^*,e_2,e_2^*\}$ is a basis. Let $D := C(B',e_3)$ . Then $e_1, e_2, e_3 \in D$ . Consider the F-circuit Y whose support is D. Then, $Y(e_i^*) = 0$ for $1 \leqslant i \leqslant 3$ and $\tilde {Y}$ is in the linear span of $\{\tilde {X_e}\}_{e\in B^*}$ by (L). Note that $D = C(B', e_3) \subseteq B \cup \{e_1, e_2, e_3\}$ . Therefore, if $e \in B^\ast $ but $e \ne e_1, e_2, e_3$ , then $e \notin D$ , and hence, $\tilde {Y}$ is in the linear span of $\tilde {X_1}, \tilde {X_2}$ , and $\tilde {X_3}$ .

Example 3.12. Let K be a field with $|K^\times |> 1$ and $\mathrm {char}(K) \neq 3$ and let $x \in K \setminus \{0,-3\}$ . We assume the trivial involution on K. Let ${\mathcal C}$ be a subset of $K^{[4]\cup [4]^*}$ consisting of the following eight vectors and their scalar multiples by nonzero elements:

$$ \begin{align*} & \begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}, \qquad \begin{pmatrix}-1 & 0 & 1 &-1 \\ 0 & 1 & 0 & 0 \end{pmatrix}, \qquad \begin{pmatrix}-1 &-1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}, \qquad \begin{pmatrix}-1 & 1 &-1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \\ & \begin{pmatrix} x & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 0 & x & 0 & 0 \\-1 & 0 & 1 &-1 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 0 &-x & 0 \\ 1 & 1 & 0 &-1 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 0 & 0 &-x \\ 1 &-1 & 1 & 0 \end{pmatrix}, \end{align*} $$

where $\begin {pmatrix} a_1&a_2&a_3&a_4 \\ b_1&b_2&b_3&b_4 \end {pmatrix}$ means $X \in K^{[4]\cup [4]^*}$ such that $X(i)=a_i$ and $X(i^*)=b_i$ with $i\in [4]$ . Then ${\mathcal C}$ is a K-signature of the orthogonal matroid whose set of bases is $\{[4], [4]^\ast \} \cup \{ijk^\ast l^\ast : ijkl = [4]\}$ . Notice that ${\mathcal C}$ satisfies (O $_3$ ) and (L-i) $'$ , but neither (O) $'$ nor (L-ii) $'$ .

We prove the following results in Section 4.

The converse of Theorem 3.14 is not true; see Example 4.22.

3.3 Orthogonal F-vector sets

Let ${\mathcal V}$ be a subset of $F^E$ . A vector $X \in {\mathcal V}$ is elementary in ${\mathcal V}$ if (i) it is nonzero, and (ii) it has a minimal support in ${\mathcal V} \setminus \{\mathbf {0}\}$ , and (iii) its support $\underline {X}$ is admissible. A transversal $T \in \mathcal {T}_n$ is a support basis of ${\mathcal V}$ if there is no $X \in {\mathcal V} \setminus \{\mathbf {0}\}$ such that $\underline {X} \subseteq T$ . A fundamental circuit form for ${\mathcal V}$ with respect to a support basis B is $\{X^{{\mathcal V}}_{B,e}: e\in B^*\}$ where $X^{{\mathcal V}}_{B,e} \in {\mathcal V}$ is such that $\mathrm {Supp}(X^{{\mathcal V}}_{B,e}) \subseteq B \triangle \{ e , e ^* \}$ and $X^{{\mathcal V}}_{B,e}(e) = 1$ . We simply write $X^{{\mathcal V}}_{B,e}$ as $X_e$ if it is clear from the context.

The axiom (V3) implies the uniqueness of the fundamental circuit form for each support basis of an orthogonal F-vector set ${\mathcal V}$ , and that every fundamental circuit form of an orthogonal F-vector set ${\mathcal V}$ consists of elementary vectors of ${\mathcal V}$ . When F is a field, a subset ${\mathcal W} \subseteq F^{[n]}$ is an F-vector set if and only if it is a linear subspace [Reference Anderson1]. We give an analogue of this for orthogonal F-vector sets.

We delay the proof of Theorem 3.16(i) to Section 4.5. Theorem 3.16(ii) can be deduced from the results of [Reference Oum22]. The condition that $\mathrm {char}(F) \neq 2$ in (ii) is crucial, since otherwise, $\mathcal {V} = \{(x,x): x \in F\}$ is a Lagrangian subspace of $F^{[1]\cup [1]^*}$ but not an orthogonal F-vector set.

3.4 Main theorems

We prove the equivalence of four notions of strong orthogonal matroids over tracts.

Similarly, we have the next two equivalences for weaker notions.

We will provide proofs for Theorems 3.18, 3.19 and 3.20 in Section 4. Since the notions of weak and strong orthogonal F-matroid coincide if F is a partial field [Reference Baker and Jin3], the tropical hyperfield ${\mathbb T}$ [Reference Rincón23], or the Krasner hyperfield ${\mathbb K}$ , it follows that the three notions of orthogonal F-matroids are equivalent when F is any of these specific tracts.

We summarize our results in Figure 2. Additionally, we remark that in Figure 2, each inclusion is strict for certain tracts; see Examples 4.22 and 4.23.

Figure 2

Summary of results in Section 3.1–3.4. In (a), we assume that $F\in \{{\mathbb T},{\mathbb K}\}$ or F is a partial field [Reference Baker and Jin3, Reference Rincón23]. In (b), we assume that F is a field with $\mathrm {char}(F) \ne 2$ .

3.5 Functoriality

Now we discuss the behavior of strong and weak orthogonal matroids over tracts with respect to tract homomorphisms. The following propositions are all straightforward from the definitions.

By the above proposition, we define the pushforward operator $f_*$ taking orthogonal $F_1$ -matroids to orthogonal $F_2$ -matroids by $f_*([\varphi ]) := [f \circ \varphi ]$ . Notice that the pushforwards are functorial: if $F_1 \xrightarrow {f} F_2 \xrightarrow {g} F_3$ are tract homomorphisms, then $(g \circ f)_\ast = g_\ast \circ f_\ast $ . If $F_2 = {\mathbb K}$ , the terminal object of the category of tracts, then the orthogonal ${\mathbb K}$ -matroid $[f \circ \varphi ]$ is the same thing as $\underline {M}_\varphi $ .

Therefore, we also have the pushforward operator $f_*$ taking orthogonal $F_1$ -signatures (resp. $F_1$ -circuit set) to orthogonal $F_2$ -signatures (resp. $F_2$ -circuit set). If $F_2 = {\mathbb K}$ , then $f_*({\mathcal C})$ is the same thing as the set of circuits of $\underline {M}_{{\mathcal C}}$ .

However, the simple pushforwards of orthogonal F-vector sets are not defined properly. In fact, let $f : F_1 \to F_2$ be a tract homomorphism respecting the involutions and let ${\mathcal V}$ be an orthogonal $F_1$ -vector set. Then the set $f_*({\mathcal V}) = \{c f(X) : c\in F_2^\times , X \in {\mathcal V}\}$ is not necessarily an orthogonal $F_2$ -vector set; see Example 4.26.

3.6 Duality

Let $\varphi : {\mathcal T}_n \to F$ be a strong Wick function over F. Its dual strong Wick function $\varphi ^\ast : {\mathcal T}_n \to F$ is defined as

$$\begin{align*}\varphi^\ast(T) := \varphi(T^\ast) \end{align*}$$

for all $T \in {\mathcal T}_n$ . It is indeed a strong Wick function with underlying orthogonal matroid $(\underline {M}_{\varphi })^\ast $ from definitions. We define the duals of weak and moderately weak Wick F-functions in the same way.

Given a strong (resp. weak) orthogonal F-signature ${\mathcal C}$ , we can define its dual strong (resp. weak) orthogonal F-signature ${\mathcal C}^\ast $ by setting

$$\begin{align*}{\mathcal C}^\ast := \{X^\ast : X \in {\mathcal C}\}, \end{align*}$$

and the underlying orthogonal matroid of ${\mathcal C}^\ast $ is $(\underline {M}_{{\mathcal C}})^\ast $ . The duals of strong and weak F-circuit sets of orthogonal matroids are defined in the same way.

3.7 Minors

Let $\varphi $ be a strong or (moderately) weak Wick function on E with coefficients in F and take $e \in E$ . Then we define $\varphi |e$ to be the function from the set of transversals of $E \setminus \{ e , e ^* \}$ to F as

$$ \begin{align*} (\varphi|e) (T) := \begin{cases} \varphi(T\cup \{e\}) & \text{if }e\text{ is nonsingular in }\underline{M}_\varphi, \\ \varphi(T \cup \{e^*\}) & \text{otherwise}. \end{cases} \end{align*} $$

We define minors of strong or weak orthogonal signatures as follows. Let $\mathcal {C}$ be a strong or weak orthogonal F-signature of an orthogonal matroid $\underline {M}$ on E. For $e\in E$ , let ${\mathcal C}|e$ be the set of functions in

$$ \begin{align*} \{\pi(X) \in F^{E \setminus \{ e , e ^* \}} : X\in \mathcal{C}\text{ with }X(e^*) = 0\text{ and }\underline{X} \neq \{e\}\} \end{align*} $$

that have minimal supports, where $\pi :F^E \to F^{E\setminus \{ e , e ^* \}}$ is the obvious projection.

The next proposition is direct from Proposition 1.16.

Minors of a strong or weak F-circuit set of an orthogonal matroid are defined in the same way as minors of an orthogonal F-signature, and an analogue of Proposition 3.26 holds.

One possible candidate of a minor of an orthogonal F-vector set ${\mathcal V} \subseteq F^E$ with respect to $e\in E$ is

$$ \begin{align*} {\mathcal V}|e := \{\pi(X) \in F^{E \setminus \{ e , e ^* \}} : X\in {\mathcal V}\text{ with }X(e^*) = 0\}, \end{align*} $$

which coincides with the deletion and the contraction of an F-vector set of a matroid in [Reference Anderson1, Section 4.2]. However, ${\mathcal V}|e$ is not necessarily an orthogonal F-signature in general, even if F is a partial field and the underlying orthogonal matroid of ${\mathcal V}$ is the lift of a matroid; see Example 4.25. We remark that if F is a field, then ${\mathcal V}|e$ is an orthogonal F-vector set by [Reference Oum22, Proposition 3.8].

3.8 Other related work

We briefly indicate how our notions of strong and weak orthogonal F-matroids generalize various flavors of orthogonal matroids in the literature, as mentioned in Section 1.

4.1 From Wick functions to orthogonal signatures

Let $\varphi $ be a weak Wick function on E with coefficients in a tract F. We denote by $\underline {M} = \underline {M}_\varphi $ the underlying orthogonal matroid of $[\varphi ]$ . We first suggest a candidate for the orthogonal signature induced from the given Wick function $\varphi $ .

Recall that the set of bases of $\underline {M}$ is $\mathrm {Supp}(\varphi ) = \{B \in {\mathcal T}_n : \varphi (B) \ne 0\}$ . For each circuit C of $\underline {M}$ , we define a function $X \in F^E$ such that $\underline {X} = C$ as follows. Let $T \supseteq C$ be a transversal such that $T \triangle \{x, x^\ast \} \in \mathrm {Supp}(\varphi )$ for all $x \in C$ , which exists by Lemma 1.15. Then for every $e, f \in C$ , we set

(4.1) $$ \begin{align} \frac{\tilde{X}(e)}{\tilde{X}(f)} = (-1)^{m^T_{e,f}} \frac{\varphi(T\triangle \{ e , e ^* \})}{\varphi(T\triangle \{ f , f ^* \})}. \end{align} $$

We call X an F-circuit of $\varphi $ with support C.

Lemma 4.1. The ratio $\frac {\tilde {X}(e)}{\tilde {X}(f)}$ is independent of the choice of T. Explicitly, let $T_1, T_2$ be distinct transversals containing C such that both $T_1 \triangle \{ x , x ^* \}$ and $T_2 \triangle \{ x , x ^* \}$ are bases of $\underline {M}$ for all $x \in C$ . Then

$$\begin{align*}(-1)^{m_1} \frac{\varphi(T_1 \triangle \{e, e^\ast\})}{\varphi(T_1 \triangle \{f, f^\ast\})} = (-1)^{m_2} \frac{\varphi(T_2 \triangle \{e, e^\ast\})}{\varphi(T_2 \triangle \{f, f^\ast\})}, \end{align*}$$

where $m_i = |T_i \cap ( \overline {e} , \overline {f} ]|$ for each $i = 1, 2$ .

Proof. We proceed by induction on $|T_1 \setminus T_2|$ . Since $T_1 \triangle \{x, x^\ast \}$ and $T_2 \triangle \{ x , x ^* \}$ with $x \in C \neq \emptyset $ are distinct bases of $\underline {M}$ , we know that $|T_1 \setminus T_2|$ is even and at least $2$ .

Suppose that $|T_1 \setminus T_2| = 2$ . Write $T_1 \setminus T_2 = \{a,b\}$ so that $T_1 \triangle T_2 = \{a, a^\ast , b, b^\ast \}$ . Then neither $T_1 \triangle \{ a , a ^* \}$ nor $T_1 \triangle \{ b , b ^* \}$ is a basis since they contain C. Thus, ${\varphi (T_1 \triangle \{ a , a ^* \}) = \varphi (T_1 \triangle \{ b , b ^* \}) = 0}$ . Denote $m = |\{\overline {a}, \overline {b}\} \cap ( \overline {e} , \overline {f} ]| $ . Note that $m_1 + m \equiv m_2\ \pmod {2}$ . By the axiom (W2) applied to $T_1 \triangle \{e, e^\ast , f, f^\ast \}$ and $T_1 \triangle \{a, a^\ast , b, b^\ast \} = T_2$ , we have

$$ \begin{align*} \varphi(T_1 \triangle \{ f , f ^* \}) \varphi(T_2 \triangle \{ e , e ^* \}) + (-1)^{m+1} \varphi(T_1 \triangle \{ e , e ^* \}) \varphi(T_2 \triangle \{ f , f ^* \}) \in N_F, \end{align*} $$

which implies the desired equality.

Now we assume that $|T_1 \setminus T_2|> 2$ . Fix $x \in C$ and let $B_i := T_i \triangle \{ x , x ^* \}$ with $i \in \{1,2\}$ . Then $B_1$ and $B_2$ are bases of $\underline {M}$ . Take $y \in T_1 \setminus T_2$ . By the symmetric exchange axiom, there is $z \in (T_1 \setminus T_2) \setminus \{y\}$ such that $B_1 \triangle \{y,y^*,z,z^*\}$ is a basis of $\underline {M}$ . Let $T_0 := T_1 \triangle \{y,y^*,z,z^*\}$ . We claim that for every $w \in C$ , the transversal $T_0 \triangle \{ w , w ^* \}$ is a basis of $\underline {M}$ . Indeed, since $T_0 \triangle \{ x , x ^* \} = B_1 \triangle \{y,y^*,z,z^*\}$ , we may assume that $w \neq x$ . Then by Proposition 1.14, $T_1 \triangle \{ w , w ^* \} = B_1 \triangle \{x, x^\ast , w, w^\ast \}$ is a basis of $\underline {M}$ . Both $B_1 \triangle \{x,x^*,y,y^*\} = T_1 \triangle \{ y , y ^* \}$ and $B_1 \triangle \{x,x^*,z,z^*\} = T_1 \triangle \{ z , z ^* \}$ contain C, and hence, neither of them is a basis. Then the symmetric exchange forces that $T_0 \triangle \{ w , w ^* \} = B_1 \triangle \{x,x^*,y,y^*,z,z^*,w,w^*\}$ is a basis. Notice that $|T_0 \triangle T_1| = 4$ and $|T_0 \triangle T_2| < |T_1 \triangle T_2|$ . Therefore, we conclude the desired equality by the claim and the induction hypothesis.

Proof. Lemma 4.1 shows the uniqueness. We assert furthermore that X is well-defined. This can be proved directly. Let T be a transversal such that $C \subseteq T$ and $T \triangle \{ x , x ^* \} \in \mathrm {Supp}(\varphi )$ for all $x\in C$ . Take $e, f, g \in C$ . Since $m_{e,f} + m_{f,g} + m_{e,g} \equiv 0\ \pmod {2}$ , we have

$$ \begin{align*} \frac{\tilde{X}(e)}{\tilde{X}(f)} \frac{\tilde{X}(f)}{\tilde{X}(g)} = (-1)^{m_{e,f}} \frac{\varphi(T\triangle \{ e , e ^* \})}{\varphi(T\triangle \{ f , f ^* \})} \cdot& (-1)^{m_{f,g}} \frac{\varphi(T\triangle \{ f , f ^* \})}{\varphi(T\triangle \{ g , g ^* \})} \\=&\ (-1)^{m_{e,g}} \frac{\varphi(T\triangle \{ e , e ^* \})}{\varphi(T\triangle \{ g , g ^* \})} = \frac{\tilde{X}(e)}{\tilde{X}(g)}.\\[-45pt] \end{align*} $$

By Proposition 4.2, the set ${\mathcal C}_\varphi $ of all projective F-circuits induced from $\varphi $ is an F-signature of $\underline {M}$ .

Proof. Let $X_1, X_2 \in \mathcal {C}_\varphi $ . We may assume that $\underline {X_1} \cap \underline {X_2}^* \neq \emptyset $ . By Lemma 1.15, there is a transversal $T_i$ containing $\underline {X_i}$ such that $T_i \triangle \{ e , e ^* \}$ is a basis for every $e \in \underline {X_i}$ and $i = 1,2$ . Note that

$$\begin{align*}\varphi(T_1 \triangle \{ e , e ^* \}) \varphi(T_2 \triangle \{ e , e ^* \}) = 0 \end{align*}$$

for all $e \in (T_1 \triangle T_2) \setminus (\underline {X_1} \triangle \underline {X_2})$ . Write $T_1 \cap T_2^* = \{e_1, e_2, \dots , e_a\}$ with $\overline {e_1} < \overline {e_2} < \dots < \overline {e_a}$ and write $\underline {X_1} \cap \underline {X_2}^* = \{e_{\alpha _1}, \dots , e_{\alpha _b}\}$ with $\alpha _1 < \dots < \alpha _b$ . Then $(T_1 \triangle T_2) \cap [n] = \{\overline {e_1},\dots ,\overline {e_a}\}$ . Let $m_{j} := |T_1 \cap ( \overline {e_{\alpha _1}} , \overline {e_{\alpha _j}} ]|$ and $n_{j} := |T_2 \cap ( \overline {e_{\alpha _1}} , \overline {e_{\alpha _j}} ]|$ for each $j \in [b]$ . Since $(T_1 \triangle T_2) \cap ( \overline {e_{\alpha _1}} , \overline {e_{\alpha _j}} ] = \{\overline {e_{k}} : \alpha _1 < k \leqslant \alpha _j\}$ , we have ${m_j+n_j} \equiv {\alpha _j - \alpha _1}\ \pmod {2}$ . By (W2) applied to $T_1$ and $T_2$ , we have

$$ \begin{align*} N_F \ni \sum_{i=1}^a (-1)^{i} \varphi(T_1 \triangle \{ e_i , e_i ^* \}) \varphi(T_2 \triangle \{ e_i , e_i ^* \}) = \sum_{i=1}^b (-1)^{\alpha_i} \varphi(T_1 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}) \varphi(T_2 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}). \end{align*} $$

Therefore,

$$ \begin{align*} \langle X_1, X_2^\ast \rangle &= \sum_{i = 1}^b \tilde{X_1}(e_{\alpha_i}) \tilde{X_2}(e_{\alpha_i}^*) \\ &= \tilde{X_1}(e_{\alpha_1}) \tilde{X_2}(e_{\alpha_1}^*) \sum_{i=1}^b \frac{\tilde{X_1}(e_{\alpha_i})}{\tilde{X_1}(e_{\alpha_1})} \frac{\tilde{X_2}(e_{\alpha_i}^*)}{\tilde{X_2}(e_{\alpha_1}^*)} \\ &= \tilde{X_1}(e_{\alpha_1}) \tilde{X_2}(e_{\alpha_1}^*) \sum_{i=1}^b (-1)^{m_i} \frac{\varphi(T_1 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \})}{\varphi(T_1 \triangle \{ e_{\alpha_1} , e_{\alpha_1} ^* \})} (-1)^{n_i} \frac{\varphi(T_2 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \})}{\varphi(T_2 \triangle \{ e_{\alpha_1} , e_{\alpha_1} ^* \})} \\ &= (-1)^{\alpha_1} \frac{\tilde{X_1}(e_{\alpha_1}) \tilde{X_2}(e_{\alpha_1}^*)}{\varphi(T_1 \triangle \{ e_{\alpha_1} , e_{\alpha_1} ^* \}) \varphi(T_2 \triangle \{ e_{\alpha_1} , e_{\alpha_1} ^* \})} \\ & \hspace{0.5cm} \times \sum_{i=1}^b (-1)^{\alpha_i} \varphi(T_1 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}) \varphi(T_2 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}) \in N_F.\\[-45pt] \end{align*} $$

4.2 From orthogonal signatures to Wick functions

Throughout this part, $\mathcal {C} \subseteq F^{E}$ is an F-signature of an ordinary orthogonal matroid $\underline {M}$ satisfying the $2$ -term orthogonality:

1. (O₂) $\langle X,Y^\ast \rangle \in N_F$ for all $X,Y \in \mathcal {C}$ with $|\underline {X} \cap \underline {Y}^*| = 2$ .

Recall that by Lemma 3.7, for each circuit C of $\underline {M}$ , the F-circuit $X\in {\mathcal C}$ (and equivalently, $\tilde {X}$ ) with $\underline {X} = C$ is unique up to multiplication by an element in $F^\times $ .

We first set $\gamma (B,B) = 1$ for every basis B of $\underline {M}$ . Let $B_1, B_2$ be two bases of $\underline {M}$ with $|B_1 \triangle B_2| = 4$ . We can write $B_1 = T \triangle \{ f , f ^* \}$ and $B_2 = T \triangle \{ e , e ^* \}$ for some transversal T containing e and f. Let $X \in \mathcal {C}$ be the F-circuit whose support $\underline {X}$ is the fundamental circuit $C(B_1, f)$ . Then $\underline {X} = C(B_2, e) \subseteq T$ , and in particular, $e,f \in \underline {X}$ . We define

$$ \begin{align*} \gamma(B_1,B_2) := (-1)^{m^T_{e,f}} \frac{\tilde{X}(e)}{\tilde{X}(f)}. \end{align*} $$

Proof. By Lemma 3.7, $\gamma (B_1,B_2)$ is independent of the choice of X for fixed T. Let $T_1 = T \triangle \{ e , e ^* \} \triangle \{ f , f ^* \} \ni e^*, f^*$ . Let $X_1\in \mathcal {C}$ be such that $\underline {X}_1 = C(B_1,e^*) = C(B_2,f^*) \ni e^*, f^*$ . It suffices to show that

$$ \begin{align*} (-1)^{m^{T}_{e,f}} \frac{\tilde{X}(e)}{\tilde{X}(f)} = (-1)^{m^{T_1}_{e,f}} \frac{\tilde{X}_1(f^*)}{\tilde{X}_1(e^*)}. \end{align*} $$

Since $(T \triangle T_1) \cap ( \overline {e} , \overline {f} ] = \max \{\overline {e},\overline {f}\}$ , we have $|m^T_{e,f}-m^{T_1}_{e,f}| = 1$ . By (O $_2$ ), $\tilde {X}(e)\tilde {X}_1(e^*) + \tilde {X}(f)\tilde {X}_1(f^\ast ) = \langle X, X_1^\ast \rangle \in N_F$ , and therefore, we obtain the desired equality.

The next lemma is obvious from the definition.

Now we define a candidate for a Wick function on E with coefficients in F whose underlying matroid is exactly $\underline {M}$ . Fix a basis $B_0$ of $\underline {M}$ , and let $\varphi _{\mathcal {C}} : \mathcal {T}_n \to F$ be such that:

1. (i) $\varphi _{\mathcal {C}}(B_0) = 1 (\not \in N_F)$ .

2. (ii) For each basis B of $\underline {M}$ other than $B_0$ , we set

   $$ \begin{align*} \varphi_{\mathcal{C}}(B) := \gamma(B', B) \varphi_{\mathcal{C}}(B'), \end{align*} $$
   where $B'$ is a basis of $\underline {M}$ such that $|B\setminus B'| = 2$ and $|B \setminus B_0| = |B' \setminus B_0| + 2$ .

3. (iii) For each non-basis transversal T, we set $\varphi (T) = 0$ .

To show that $\varphi _{\mathcal {C}}$ is well-defined, we need a result on the basis graphs of the orthogonal matroids.

The basis graph $\Gamma _{\underline {N}}$ of an ordinary orthogonal matroid $\underline {N}$ is a graph whose vertex set is the set of bases of $\underline {N}$ , and two vertices $B_1$ and $B_2$ are adjacent if and only if $|B_1 \setminus B_2| = 2$ . For every graph G, a directed cycle C of length $\ell \geqslant 2$ is a sequence $(v_0,v_1)$ , $(v_1,v_2)$ , $\cdots $ , $(v_{\ell -1},v_{\ell })$ of ordered pairs of adjacent vertices in G such that all $v_k$ are distinct except for $v_0 = v_\ell $ . We simply write C as a sequence $v_0,v_1,\ldots ,v_{\ell -1},v_0$ of vertices. We denote by $-C$ the directed cycle $v_0,v_{\ell -1},\dots ,v_1,v_0$ . For directed cycles $C_0,C_1,\dots ,C_m$ of G, we say $C_0$ is generated by $C_1,\dots ,C_m$ if for all vertices $u,v$ in G, two ordered pairs $(u,v)$ and $(v,u)$ appear the same number of times in $-C_0$ , $C_1$ , $\cdots $ , $C_m$ ; see Figure 3.

Figure 3

$C_0$ is generated by $C_1, C_2, C_3$ .

The following theorem generalizes Maurer’s Homotopy Theorem for matroids [Reference Maurer20].

Proof. (i) If $B_1,B_2,B_3$ are bases of $\underline {M}$ with $|B_i\setminus B_j| = 2$ for all distinct $i,j \in [3]$ , then there exist a transversal T and distinct elements $e_1^*,e_2^*,e_3^* \in T$ such that $B_i = T \triangle \{ e_i , e_i ^* \} \ni e_i$ for each $i\in [3]$ . For distinct $i,j \in [3]$ , let $T_{ij} := T \triangle \{ e_i , e_i ^* \} \triangle \{ e_j , e_j ^* \}$ and $X_{ij} \in \mathcal {C}$ be the F-circuit with $\underline {X_{ij}} = C(B_i, e_j) = C(B_j, e_i) \subseteq T_{ij}$ . Then

$$ \begin{align*} \gamma(B_i,B_j) = (-1)^{m_{ij}} \frac{\tilde{X}_{ij}(e_i)}{\tilde{X}_{ij}(e_j)}, \end{align*} $$

where $m_{ij} := |T_{ij} \cap ( \overline {e_i} , \overline {e_j} ]|$ . Let Y be the F-circuit with $\underline {Y} = C(B_1, e_1^*) \subseteq T$ . Then $\underline {Y} = C(B_2, e_2^*)= C(B_3, e_3^*)$ , and thus, $\{e_1^*,e_2^*,e_3^*\} \subseteq \underline {Y}$ . By (O $_2$ ), if $i \ne j \in [3]$ , we have that $\tilde {X}_{ij}(e_i)\tilde {Y}(e_i^*) + \tilde {X}_{ij}(e_j)\tilde {Y}(e_j^*) = \langle X_{ij}, Y^\ast \rangle \in N_F$ . Thus,

$$ \begin{align*} \frac{\tilde{X}_{12}(e_1)}{\tilde{X}_{12}(e_2)} \frac{\tilde{X}_{23}(e_2)}{\tilde{X}_{23}(e_3)} \frac{\tilde{X}_{31}(e_3)}{\tilde{X}_{31}(e_1)} = \left( - \frac{\tilde{Y}(e_2^*)}{\tilde{Y}(e_1^*)} \right) \left( - \frac{\tilde{Y}(e_3^*)}{\tilde{Y}(e_2^*)} \right) \left( - \frac{\tilde{Y}(e_1^*)}{\tilde{Y}(e_3^*)} \right) = -1. \end{align*} $$

By relabeling, we may assume that $\overline {e_1}<\overline {e_2}<\overline {e_3}$ . Then $(T_{12} \cap ( \overline {e_1} , \overline {e_2} ]) \triangle (T_{23} \cap ( \overline {e_2} , \overline {e_3} ]) \triangle (T_{13} \cap ( \overline {e_1} , \overline {e_3} ]) = \{\overline {e_{2}}\}$ . Hence, $m_{12} + m_{23} + m_{13}$ is odd, and therefore, $\gamma (B_1,B_2) \gamma (B_2,B_3) \gamma (B_3,B_1) = 1$ .

(ii) By (i) and Lemma 4.6, we may assume that the directed cycle $B_1, B_2, B_3, B_4, B_1$ is not generated by directed cycles of length $3$ . Then $|B_i\setminus B_{i+1}| = 2$ and $|B_i \setminus B_{i+2}| = 4$ for all $i\in [4]$ , where all subscripts are read modulo $4$ . Thus, there exist a transversal T and distinct elements $e_1,e_2,e_3,e_4 \in T$ such that $B_1 = T_{12}$ , $B_2 = T_{13}$ , $B_3 = T_{34}$ , and $B_4 = T_{24}$ , where $T_{I} = T \triangle \bigcup _{i\in I }\{ e_i , e_i ^* \}$ for all $I \subseteq [4]$ . In addition, none of T, $T_{14}$ , $T_{23}$ , and $T_{1234}$ is a basis.

Let $X_1, X_3, Y_3, Y_1 \in \mathcal {C}$ be F-circuits such that $\underline {X_i} \subseteq T_i$ and $\underline {Y_{4-i}} \subseteq T_{2i4}$ for each $i \in \{1,3\}$ . Then

$$ \begin{align*} \gamma(T_{12},T_{13}) = (-1)^{m_1} \frac{\tilde{X}_1(e_3)}{\tilde{X}_1(e_2)}, \\ \gamma(T_{13},T_{34}) = (-1)^{m_3} \frac{\tilde{X}_3(e_4)}{\tilde{X}_3(e_1)}, \\ \gamma(T_{12},T_{24}) = (-1)^{n_3} \frac{\tilde{Y}_3(e_1^*)}{\tilde{Y}_3(e_4^*)}, \\ \gamma(T_{24},T_{34}) = (-1)^{n_1} \frac{\tilde{Y}_1(e_2^*)}{\tilde{Y}_1(e_3^*)}, \end{align*} $$

where $m_1 := |T_1 \cap ( \overline {e_2} , \overline {e_3} ]|$ , $m_3 := |T_3 \cap ( \overline {e_1} , \overline {e_4} ]|$ , $n_3 := |T_{124} \cap ( \overline {e_1} , \overline {e_4} ]|$ , and $n_1 := |T_{234} \cap ( \overline {e_2} , \overline {e_3} ]|$ . Note that $m_1+m_3+n_3+n_1$ is even, since $(T_1 \cap ( \overline {e_2} , \overline {e_3} ]) \triangle (T_{234} \cap ( \overline {e_2} , \overline {e_3} ]) =\{ \overline {e_1}, \overline {e_2}, \overline {e_3}, \overline {e_4}\} \cap ( \overline {e_2} , \overline {e_3} ]$ , and $(T_3 \cap ( \overline {e_1} , \overline {e_4} ]) \triangle (T_{124} \cap ( \overline {e_1} , \overline {e_4} ]) = \{ \overline {e_1},\overline {e_2},\overline {e_3},\overline {e_4}\} \cap ( \overline {e_1} , \overline {e_4} ]$ .

Suppose for contradiction that $X_1(e_1^*) \neq 0$ (i.e., $e_1^* \in \underline {X_1}$ ). Notice that $\underline {X_1} = C(B_1,e_2)$ is the unique circuit of $\underline {M}$ contained in $T_1$ , and the subtransversal $T_1 \setminus \{e_1^*\}$ is independent. Since $T_1$ is not a basis, $T = (T_1 \setminus \{e_1^*\}) \cup \{e_1\}$ is a basis, a contradiction. Thus, $X_1(e_1^*) = 0$ . Similarly, one can check that all of $X_1(e_4)$ , $X_3(e_2)$ , $X_3(e_3^*)$ , $Y_3(e_2^*)$ , $Y_3(e_3)$ , $Y_1(e_1)$ , and $Y_1(e_4^*)$ are zero, because none of T, $T_{14}$ , $T_{23}$ and $T_{1234}$ is a basis of $\underline {M}$ . Therefore, by (O $_2$ ), we have

$$ \begin{align*} \tilde{X}_1(e_2)\tilde{Y}_1(e_2^*) + \tilde{X}_1(e_3)\tilde{Y}_1(e_3^*) = \langle X_1, Y_1^\ast \rangle \in N_F, \end{align*} $$

and

$$ \begin{align*} \tilde{X}_3(e_1)\tilde{X}_3(e_1^*) + \tilde{X}_3(e_4)\tilde{Y}_3(e_4^*) = \langle X_3, Y_3^\ast \rangle \in N_F. \end{align*} $$

Therefore,

$$ \begin{align*} \gamma(T_{12},T_{13}) \gamma(T_{13},T_{34}) = (-1)^{m_1+m_3} \frac{\tilde{X}_1(e_3)}{\tilde{X}_1(e_2)} \frac{\tilde{X}_3(e_4)}{\tilde{X}_3(e_1)} = (-1)^{n_1+n_3} \frac{\tilde{Y}_1(e_2^*)}{\tilde{Y}_1(e_3^*)} \frac{\tilde{Y}_3(e_1^*)}{\tilde{Y}_3(e_4^*)} = \gamma(T_{24},T_{34}) \gamma(T_{12},T_{24}). \end{align*} $$

By Lemma 4.6, we obtain that

$$ \begin{align*} &\gamma(B_1,B_2) \gamma(B_2,B_3) \gamma(B_3,B_4) \gamma(B_4,B_1) \\ &= \gamma(T_{12},T_{13}) \gamma(T_{13},T_{34}) \gamma(T_{24},T_{34})^{-1}\gamma(T_{12},T_{24})^{-1} = 1.\\[-37pt] \end{align*} $$

Proof. It suffices to show that for arbitrary paths $P = B_0 B_1 \dots B_k$ and $P' = B_0' B_1' \dots B_\ell '$ in $\Gamma _{\underline {M}}$ , if $B_0 = B_0'$ and $B_k = B_\ell '$ , then

$$ \begin{align*} \prod_{i=0}^{k-1} \gamma(B_{i},B_{i+1}) = \prod_{j=0}^{\ell-1} \gamma(B_{j},B_{j+1}). \end{align*} $$

This is straightforward from Lemmas 4.6, 4.8, and Theorem 4.7.

Proof. We only need to prove (W2). Take $T_1,T_2 \in \mathcal {T}_n$ with $T_1 \cap T_2^* = \{e_1,\dots ,e_a\}$ , where $\overline {e_1}<\dots <\overline {e_a}$ . If $T_1$ is a basis of $\underline {M}$ , then $\varphi (T_1 \triangle \{ i , i ^* \}) = 0$ for all $i\in [n]$ , and thus, $\sum _{i=1}^a (-1)^i \varphi (T_1 \triangle \{ e_i , e_i ^* \}) \varphi (T_2 \triangle \{ e_i , e_i ^* \}) \in N_F$ . Therefore, we may assume that $T_1$ is not a basis, and similarly, we may assume that $T_2$ is not a basis. Then there exist $X_1, X_2 \in \mathcal {C}$ such that $\underline {X_i} \subseteq T_i$ for $i = 1, 2$ . Write $\underline {X_1} \cap \underline {X_2}^* = \{e_{\alpha _1}, \dots , e_{\alpha _b}\}$ with $\alpha _1 < \dots < \alpha _b$ . For $i \in [a] \setminus \{\alpha _1,\dots ,\alpha _b\}$ , at least one of $T_1 \triangle \{ e_i , e_i ^* \}$ and $T_2 \triangle \{ e_i , e_i ^* \}$ is not a basis. Hence,

$$ \begin{align*} \sum_{i=1}^a (-1)^i \varphi(T_1 \triangle \{ e_i , e_i ^* \}) \varphi(T_2 \triangle \{ e_i , e_i ^* \}) = \sum_{i=1}^b (-1)^{\alpha_i} \varphi(T_1 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}) \varphi(T_2 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}). \end{align*} $$

Therefore, we may assume that $b \geqslant 1$ . We can also assume that there exists $c\in [b]$ such that both $B_1 := T_1 \triangle \{ e_{\alpha _c} , e_{\alpha _c} ^* \}$ and $B_2 := T_2 \triangle \{ e_{\alpha _c} , e_{\alpha _c} ^* \}$ are bases. Then $\underline {X_1} = C(T_1 \triangle \{ e_{\alpha _c} , e_{\alpha _c} ^* \}, e_{\alpha _c})$ and $\underline {X_2} = C(T_2 \triangle \{ e_{\alpha _c} , e_{\alpha _c} ^* \}, e_{\alpha _c}^*)$ , and therefore, $T_j \triangle \{ e_{\alpha _i} , e_{\alpha _i} ^* \}$ is a basis for each $i\in [b]$ and $j = 1, 2$ .

For each $i\in [b]$ , let $m_i := |T_1 \cap ( \overline {e_{\alpha _c}} , \overline {e_{\alpha _i}} ]|$ and $n_i := |T_2 \cap ( \overline {e_{\alpha _c}} , \overline {e_{\alpha _i}} ]|$ . By the definition of $\varphi _{\mathcal {C}}$ , we have

$$ \begin{align*} \frac{\tilde{X}_1(e_{\alpha_i})}{\tilde{X}_1(e_{\alpha_c})} = (-1)^{m_i} \frac{\varphi(T_1\triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \})}{\varphi(T_1\triangle \{ e_{\alpha_c} , e_{\alpha_c} ^* \})} \quad\text{and}\quad \frac{\tilde{X}_2(e_{\alpha_i}^*)}{\tilde{X}_2(e_{\alpha_c}^*)} = (-1)^{n_i} \frac{\varphi(T_2\triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \})}{\varphi(T_2\triangle \{ e_{\alpha_c} , e_{\alpha_c} ^* \})}. \end{align*} $$

Since $(T_1 \triangle T_2) \cap ( \overline {e_{\alpha _c}} , \overline {e_{\alpha _i}} ]$ equals $\{\overline {e_{k}} : \alpha _c < k \leqslant \alpha _i\}$ if $c<i$ and $\{\overline {e_{k}} : \alpha _i < k \leqslant \alpha _c\}$ otherwise, we have $m_i + n_i \equiv \alpha _i - \alpha _c\ \pmod {2}$ . By the orthogonality relation (O), we have

$$ \begin{align*} \sum_{i=1}^b \tilde{X}_1(e_{\alpha_i}) \tilde{X}_2(e_{\alpha_i}^*) = \langle X_1, X_2^\ast \rangle \in N_F. \end{align*} $$

Therefore,

$$ \begin{align*} &\sum_{i=1}^b (-1)^{\alpha_i} \varphi(T_1 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}) \varphi(T_2 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \})\\ &= (-1)^{\alpha_c} \sum_{i=1}^b (-1)^{m_i+n_i} \varphi(T_1 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}) \varphi(T_2 \triangle \{ e_{\alpha_i} , e_{\alpha_i} ^* \}) \\ &= (-1)^{\alpha_c} \frac{\varphi(T_1 \triangle \{ e_{\alpha_c} , e_{\alpha_c} ^* \})}{\tilde{X}_1(e_{\alpha_c})} \frac{\varphi(T_2 \triangle \{ e_{\alpha_c} , e_{\alpha_c} ^* \})}{\tilde{X}_2(e_{\alpha_c}^*)} \sum_{i=1}^b \tilde{X}_1(e_{\alpha_i}) \tilde{X}_2(e_{\alpha_i}^*) \in N_F.\\[-45pt] \end{align*} $$