**Maple Worksheets created with CLIFFORD/Bigebra**

The worksheets posted below have been created with Maple 13 Classic Interface and CLIFFORD package ver. 12 to verify results proven in the following three-part paper:

- [1] R. Ablamowicz and B. Fauser,
*On the Transposition Anti-Involution in Real Clifford Algebras I: The Transposition Map*, 2010 (to appear in Linear and Multilinear Algebra, DOI: 10.1080/03081087.2010.517201) - [2] R. Ablamowicz and B. Fauser,
*On the Transposition Anti-Involution in Real Clifford Algebras II: Stabilizer Groups of Primitive Idempotents,*2010 (to appear in Linear and Multilinear Algebra, DOI: 10.1080/03081087.2010.517202) - [3] R. Ablamowicz and B. Fauser,
*On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces*, 2011 (submitted to Linear and Multilinear Algebra)

The CLIFFORD package is a free Maple add-on package which can be downloaded from http://math.tntech.edu/rafal/ for various versions of Maple Computer Algebra System. To run these worksheets, you must install CLIFFORD ver. 12 (or later) according to the posted instructions. Ver. 12 of CLIFFORD runs under Maple 12, 13, 14, 15, and 16. This version 12 will be updated to Maple 13, 14, 15, and 16 in due course after they are thoroughly tested. Each worksheet also can be viewed as a pdf file. A brief description of each worksheet is included in the listing below.

NOTE: To open and execute Maple worksheets displayed below, use this CLIFFORD library Cliffordlib_M12_06192011.zip Also, download Walshpackage.m file with our new non-standard supplementary package "Walshpackage" for computation of the Clifford product in CL(p,q) with Walsh functions. This package is not yet included in the standard CLIFFORD distribution.

- transversal.mws, transversal.pdf [7 pages] - this worksheet checks three conjectures for simple and semisimple Clifford algebras proven in paper [1] above.
- Conjecture 1: data[7] is a transversal of the stabilizer Gp,q(f) of a primitive idempotent f in Gp,q
- Conjecture 2: data[6] is a transversal of Tp,q(f) in Gp,q
- Conjecture 3: data[5] is a transversal of Tp,q(f) in Gp,q(f)

- diagrams252627.mws, diagrams252627.pdf [39 pages] - this worksheet verifies four commutative diagrams in paper [1]: two diagrams (25) on page 12, diagram (26) on page 13, and diagram (27) on page 15. It uses an experimental "Walshpackage" which uses Walsh functions to compute Clifford product of two basis monomials m1 and m2 in signature (p,q) (see Lounesto's Chapt.21 in "Clifford Algebras and Spinors", 2nd ed., Cambridge, 2001. A procedure which implements this product on the monomial level is called cmulWalsh3. It is used as cmulWalsh3(m1,m2,[p,q]).
- Walsh3.mws, Walsh3.pdf [50 pages] - Walsh3.mws contains Maple code of "Walshpackage" with examples and time comparisons between cmulWalsh3, cmulRS, and cmulNUM procedures while Walsh3.pdf is a pdf version of Walsh3.mws

- reciprocal_basis.mws, reciprocal_basis.pdf [47 pages] - this worksheet verifies a variety of results and computes Examples 1-5 from paper [1]:
- it computes reciprocal bases using $T_\epsilon$ (defined as a Maple procedure EPS) and $T_\epsilon^\tilde$ (defined as a Maple procedure tp)
- verifies that tp and EPS are isomorphisms of /\ V and Cl(p,q) as vector spaces
- checks that tp is an anti-involution of /\V
- checks that tp = reversion in Cl(p,0) and tp = conjugation in Cl(0,q)
- shows that tp is a Clifford map
- verifies that tp = reversion o EPS = EPS o reversion
- shows that a procedure rbas which uses tp computes a dual basis to the Grassmann basis in Cl(p,q)
- defines Lounesto's inner product on Cl(p,q) as a procedure LIP
- checks formulas in Section 4 of [1]
- verifies results in Proposition 2 in [1]
- computes Examples 1--5 displayed in Section 5 in [1]

- real_simple.mws, real_simple.pdf [71 pages] - this worksheet verifies assertions made for real simple Clifford algebras Cl(p,q) when p - q <> 1 mod 4 and p - q = 0,1,2 mod 8 in paper [2]. Thus, signatures tested are:
(1,1), (2,0), (2,2), (3,1), (0,6), (3,3), (4,2), (0,8), (1,7), (4,4), (5,3), and (8,0)
- verifying that the basis vectors e1, e2,..., en in Cl(p,q) are represented by either symmetric or antisymmetric matrices
- finding matrix [u] for a general element u in Cl(p,q) in spinor representation and verifying that [tp(u)] = transpose([u])
- verifying Lemma 1 in [2]
- action of tp on spinors and primitive idempotents
- finding a new transposition scalar form Lambda(Psispinor,Phispinor) = cmul(tp(Psispinor),Phispinor) = lambda*f for some lambda in K = fCl(p,q)f
- creating and saving data file LAMBDA.m which stores values of the Lambda form
- comparing the Lambda form with beta_plus and beta_minus
- creating data files BETA_PLUS.m and BETA_MINUS.m
- action of tp on idempotents
- using Stab.m data file to compute stabilizer groups Gp,q(f) saved later a data file StabilizerGroup.m
- verifying Propositions 1, 3, 4, 5 in [2] in the real simple case
- finding orders of elements in Gp,q(f) and finding generators for these groups displayed in Table 1 in [2]
- storing generators of the stabilizer groups Gp,q(f) in a data file Gen.m
- applying tp to spinors
- verifying Lemma 2 in [1]
- preparing Example for [2]

- complex_simple.mws, complex_simple.pdf [169 pages] - this worksheet verifies assertions made for complex simple Clifford algebras Cl(p,q) when p - q <> 1 mod 4 and p - q = 3,7 mod 8 in paper [2]. Thus, signatures tested are:
(1,2), (3,0), (0,5), (2,3), (4,1), (1,6), (3,4), (5,2), (7,0), (0,9), (2,7), (4,5), (6,3) and (8,1)
- computations from real_simple.mws are repeated for complex simple Clifford algebras Cl(p,q) including a verification that Gp,q < Lip(p,q) and Gp,q < Pin(p,q) where Lip(p,q) is the Lipschitz group
- computing data for Table 2 in [2]

- quaternionic_simple.mws, quaternionic_simple.pdf [74 pages] - this worksheet verifies assertions made for quaternionic simple Clifford algebras Cl(p,q) when p - q <> 1 mod 4 and p - q = 4,5,6 mod 8 in paper [2]. Thus, signatures tested are:
(0,4), (1,3), (1,5), (2,4), (2,6), (3,5), (4,0), (5,1), (6,0), (6,2), and (7,1)
- repeating computations from real simple case in quaternionic simple case
- showing that tp in spinor representation gives quaternionic Hermitian conjugation while in the left regular representation tp just gives the ordinary matrix transposition
- Example 5 in [2]
- properties of tp in the quaternionic simple case
- action of tp on primitive idempotents
- stabilizers of primitive idempotents
- data for Table 3 in [2]
- finding cosets of Gp,q(f) in Gp,q

- real_semisimple.mws, real_semisimple.pdf [62 pages] - this worksheet verifies assertions made for real semisimple Clifford algebras Cl(p,q) when p - q = 1 mod 4 and p - q = 0,1,2 mod 8 in paper [2]. Thus, signatures tested are:
(2,1), (3,2), (0,7), (4,3), (1,8), (5,4), and (9,0)
- essentially this worksheets extends computation from the worksheet real_simple to the real semisimple signatures
- verifying properties of tp in Lemma 1 in [2]
- acting with tp on spinors in Cl(p,q)f + Cl(p,q) fg and idempotents f, fg where fg = grade involuted f
- computation of Lambda spinor product which involves tp in the real semisimple Clifford algebras
- finding stabilizers of primitive idempotents
- verifying Proposition 6 in [2] in real semisimple case
- finding and saving generators of Gp,q(f) which are later displayed in Table 4 in [2]
- computation of orbits of f an fg for Proposition 6 in [2]
- computing Example 7 in [2]

- quaternionic_semisimple.mws, quaternionic_semisimple.pdf [95 pages] - this worksheet verifies assertions made for quaternionic semisimple Clifford algebras Cl(p,q) when p - q = 1 mod 4 and p - q = 4,5,6 mod 8 in paper [2]. Thus, signatures tested are:
(0,3), (1,4), (2,5), (3,6), (5,0), (6,1), and (7,2)
- computing Table 4 for [2]
- showing that tp in spinor representation gives quaternionic Hermitian conjugation while in the left regular representation it gives the ordinary matrix transposition
- explaining why matrices in spinor representation of the basis vectors e1, e2, ..., en, are either quaternionic hermitian when ei^2 = 1 or ei = tp(ei), i=1,.,,.p, or quaternionic anti-hermitian when ei^2 = -1 or ei = -tp(ei), for i=p+1,...,n=p+q,
- showing that [tp(u)] = [u]^{qh} where qh stands for the quaternionic Hermitian conjugation for any u in Cl(p,q) in spinor representation
- verifying Lemma 1 in [2] in quaternionic semisimple Clifford algebras Cl(p,q)
- checking properties of tp in quaternionic semisimple Clifford algebras Cl(p,q)
- action of tp on spinors and half-spinors
- computing and saving the transposition scalar product, and comparing it with beta_plus and beta_minus
- action of tp on primitive idempotents f and fg
- finding two different orbits of f and fg
- computing and saving stabilizers of f and fg. Of course, Gp,q(f) = Gp,q(fg)
- finding and storing generators of the stabilizers for Table 5 in [2]

- F3groups.mws, F3groups.pdf [14 pages] - this worksheet verifies assertions about a group F3 made in [2], in particular, in Table 3 in [2]
- data_files.mws, data_files.pdf [102 pages] - This worksheet summarizes and allows one to display data that has been collected in the worksheets real_simple.mws, complex_simple, quaternionic_simple, real_semisimple, and quaternionic_semisimple.mws shown above, and in tables LAMBDA (Maple file Lambda.m), StabilizerGroup (Maple file StabilizerGroup.m), Gen (Maple file Gen.m), G (Maple file G.m), and Stab (Maple file Stab.m) displayed below.
- it includes a verification that Gpq,(f) is normal in Gp,q for all signatures
- it displays and collects data for Tables 1,2,3,4,5 in [2]

- conjecturesGpqf.mws, conjecturesGpqf.pdf [77 pages] - This worksheet allows us to check conjectures about the structure of the stabilizer Gp,q(f) of a primitive idempotent f and its subgroups Kp,q(f) and Tp,q(f) defined in [3]. In particular,
- it computes orders of Gp,q, Gp,q(f), Kp,q(f), Tp,q(f) for any signature (p,q) and p+q <= 9,
- it verifies conjectures in Lemma 1, Theorem 1, and Corollary 2 in [3] and results stated in [2] in the group order of Gp,q(f):
- Conjecture 0: The groups Gp,q(f), Tp,q(f), and Kp,q(f) are normal in Gpq,
- Conjecture 1: Elements of Tp,q(f) and Kp,q(f) commute,
- Conjecture 2: The intersection Tpqf \cap Kpqf = G'pq = {Id,-Id} which is the commutator subgroup of Gp,q,
- Conjecture 3: Gp,q(f) = Tp,q(f) Kp,q(f),
- Conjecture 4: Tp,q(f) is normal in Gp,q(f) and Kp,q(f) is normal in Gp,q(f),
- Conjecture 5: Relation among orders |Gp,q(f)|, |Tp,q(f)|, and |Kp,q(f)| of groups Gp,q(f), Tp,q(f), and Kp,q(f) (see Theorem 1 in [3]),
- Conjecture 6:
- Part A: (Gpqf/Kpqf) / (Tpqf/Kpqf) = Gpqf/Tpqf IS TRUE ONLY IN REAL CASE and the transversal of Tpqf in Gpqf spans K = fCL(p,q)f = data[6]. NOTE: To make Part A true in each case, we really need (Gpqf/ Hpqf ) / (Tpqf/Hpqf) = Gpqf/Tpqf where Hpqf = Tpqf \cap Kpqf = {+/- Id} = G'pq.
- Part B: (Gpqf/G'pq) / (Tpqf/G'pq) = Gpqf/Tpqf and the transversal of Tpqf in Gpqf spans K = fCL(p,q)f = data[6]. BOTH PARTS OF B ARE ALWAYS TRUE

- Conjecture 7 = Conjecture 1 in the worksheet transversal.mws: data[7] = transversal of Gp,q(f) in Gp,q,
- Conjecture 8 = Conjecture 3 in the worksheet transversal.mws: (Gp,q/Tp,q(f)) / (Gp,q(f)/Tp,q(f)) = Gp,q/Gp,q(f) and transversal of Tp,q(f) in Gp,q is data[5],
- Conjecture 9:
- (i) For every x in Tp,q(f), the stabilizer Gp,q(f) is a subgroup of the centralizer C_Gp,q(x) of x in Gp,q.
- (ii) The stabilizer Gp,q(f) = Intersection of all centralizers C_Gp,q(x) of x in Gp,q (intersection taken over all elements x in Tp,q(f) = "all elements in G which commute with every element in Tp,q(f)" = Centralizer of Tp,q(f) in Gp,q.

- Conjecture 10 (The Second Isomorphism Theorem):
- (a) Gpqf / Kpqf = Tpqf/G'pq
- (b) Gpqf / Tpqf = Kpqf/G'pq

- BETA_PLUS_MINUS.mws, BETA_PLUS_MINUS.pdf [73 pages] - this worksheet computes and saves all three scalar products on spinor spaces: the new Lambda product defined as Lambda(Psispinor,Phispinor) = cmul(tp(Psispinor),Phispinor); the beta_plus and the beta_minus products shown in display (44) in [3] following Lounesto's definition. In particular,
for all simple and semisimple Clifford algebras Cl(p,q), p+q <= 9,
- This worksheet computes the beta_plus form and saves it as BETA_PLUS.m data file;
- This worksheet computes the beta_minus form and saves it as BETA_MINUS.m data file;
- This worksheet computes the Lambda form introduced in [2] and discussed at length in [3]. It saves the form in a data file LAMBDA.m;
- A comparison is made between the Lambda form and the beta forms;
- Invariance groups are found for all Lambda, beta_plus, and beta_minus forms: The invariance groups of the beta forms are known from Lounesto's Tables 1 and 2 on page 236 in ``Clifford Algebras and Spinors", 2nd ed., Cambridge U. Press, 2001.

- Maple data files (require Maple to view their content):
- LAMBDA.m - contains the new transposition scalar product Lambda
- BETA_PLUS.m - contains the scalar product beta_plus
- BETA_MINUS.m - contains the scalar product beta_minus
- F3.m - contains the group F3
- StabilizerGroup.m - contains the stabilizer groups Gp,q(f) of a default primitive idempotent f
- Walshpackage.m - contains an experimental 'Walshpackage" for computation of the Clifford product in CL(p,q) with Walsh functions. Check Chapter 21 in P. Lounesto's "Clifford Algebras and Spinors", 2nd. ed., Cambridge, 2001.

Last revised: June 17, 2012