Skip to content

Commit 2bf2269

Browse files
jw-allenjw-allen
authored
Merge pull request #16 from jw-allen/v0.6.3
v0.6.3
2 parents 9abc40c + 6f0c2b1 commit 2bf2269

18 files changed

+2430
-55
lines changed

‎.gitignore

-1
Original file line numberDiff line numberDiff line change
@@ -19,7 +19,6 @@ doc/*.six
1919
doc/*.tex
2020
doc/*.toc
2121
doc/*.txt
22-
doc/*.xml
2322

2423
**.pdf
2524

‎README

+3-3
Original file line numberDiff line numberDiff line change
@@ -1,6 +1,6 @@
11
+--------------------------------------------------------+--------------------+
22
| | Copyright (C) 2020 |
3-
| SBStrips (v0.6.2) | Joe Allen |
3+
| SBStrips (v0.6.3) | Joe Allen |
44
| | [he/him/his] |
55
+--------------------------------------------------------+--------------------+
66

@@ -28,9 +28,9 @@ The SBStrips can be installed by following these steps (assuming you already
2828
have GAP installed.)
2929

3030
1. Download the archive file
31-
sbstrips/archive/v0.6.2.tar.gz
31+
sbstrips/archive/v0.6.3.tar.gz
3232
from
33-
https://github.com/jw-allen/sbstrips/archive/v0.6.2.tar.gz
33+
https://github.com/jw-allen/sbstrips/archive/v0.6.3.tar.gz
3434

3535
2. Unpack it into your pkg directory. It should create a subdirectory
3636
called sbstrips.

‎doc/ApExampleSbas.xml

+247
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,247 @@
1+
<#GAPDoc Label="AppExampleAlgebras">
2+
3+
<Section>
4+
<Heading>
5+
The function
6+
</Heading>
7+
For your convenience, &SBStrips; comes bundled with <M>5</M> SB algebras
8+
built in. We detail these algebras in this appendix. They may be obtained by
9+
calling <Ref Func="SBStripsExampleAlgebra"/>. <P />
10+
11+
<ManSection>
12+
<Func Name="SBStripsExampleAlgebra" Arg="n"/>
13+
<Description>
14+
Arguments: <A>n</A>, an integer between <C>1</C> and <C>5</C> inclusive
15+
<Br />
16+
</Description>
17+
<Returns>
18+
a SB algebra
19+
<P />
20+
</Returns>
21+
<Description>
22+
Calling this function with argument <C>1</C>, <C>2</C>, <C>3</C>,
23+
<C>4</C> or <C>5</C> respectively returns the algebras described in
24+
subsections <Ref Subsect="ExAlg1"/>, <Ref Subsect="ExAlg2"/>, <Ref
25+
Subsect="ExAlg3"/>, <Ref Subsect="ExAlg4"/> or <Ref Subsect="ExAlg5"/>.
26+
</Description>
27+
</ManSection>
28+
</Section>
29+
30+
<Section>
31+
<Heading>
32+
The algebras
33+
</Heading>
34+
35+
Each algebra is of the form <M>KQ/\langle \rho \rangle</M>, where <M>K</M>
36+
is the field <C>Rationals</C> in &GAP; and where <M>Q</M> and <M>\rho</M> are
37+
respectively a quiver and a set of relations. These change from example to
38+
example.
39+
<P />
40+
41+
The &LaTeX; version of this documentation provides pictures of each quiver.
42+
43+
<Subsection Label="ExAlg1">
44+
<Heading>
45+
Algebra <M>1</M>
46+
</Heading>
47+
48+
The quiver and relations of this algebra are specified to &QPA; as follows.
49+
<Example><![CDATA[
50+
gap> quiv := Quiver(
51+
> 2,
52+
> [ [ 1, 1, "a" ], [ 1, 2, "b" ], [ 2, 1, "c" ], [ 2, 2, "d" ] ]
53+
> );
54+
<quiver with 2 vertices and 4 arrows>
55+
pa := PathAlgebra( Rationals, quiv );
56+
<Rationals[<quiver with 2 vertices and 4 arrows>]>
57+
gap> rels := [
58+
> pa.a * pa.a, pa.b * pa.d, pa.c * pa.b, pa.d * pa.c,
59+
> pa.c * pa.a * pa.b, (pa.d)^4,
60+
> pa.a * pa.b * pa.c - pa.b * pa.c * pa.a
61+
> ];
62+
[ (1)*a^2, (1)*b*d, (1)*c*b, (1)*d*c, (1)*c*a*b,
63+
(1)*d^4, (1)*a*b*c+(-1)*b*c*a ]
64+
]]></Example>
65+
<Alt Only="LaTeX">
66+
Here is a picture of the quiver.
67+
<Display>
68+
\begin{tikzcd}
69+
1 \ar[loop left, "a"] \ar[r, bend left, "b"]
70+
&amp; 2 \ar[l, bend left, "c"] \ar[loop right, "d"]
71+
\end{tikzcd}
72+
</Display>
73+
<P />
74+
</Alt>
75+
The relations of this algebra are chosen so that the nonzero paths of
76+
length <M>2</M> are: <C>a*b</C>, <C>b*c</C>, <C>c*a</C>, <C>d*d</C>.
77+
<P />
78+
79+
The simple module associated to vertex <C>v2</C> has infinite syzygy type.
80+
</Subsection>
81+
82+
<Subsection Label="ExAlg2">
83+
<Heading>
84+
Algebra <M>2</M>
85+
</Heading>
86+
87+
The quiver and relations of this algebra are specified to &QPA; as follows.
88+
<Example><![CDATA[
89+
gap> quiv := Quiver(
90+
> 3,
91+
> [ [ 1, 2, "a" ], [ 2, 3, "b" ], [ 3, 1, "c" ] ]
92+
> );
93+
<quiver with 3 vertices and 3 arrows>
94+
gap> pa := PathAlgebra( Rationals, quiv );
95+
<Rationals[<quiver with 3 vertices and 3 arrows>]>
96+
gap> rels := NthPowerOfArrowIdeal( pa, 4 );
97+
[ (1)*a*b*c*a, (1)*b*c*a*b, (1)*c*a*b*c ]
98+
]]></Example>
99+
<Alt Only="LaTeX">
100+
Here is a picture of the quiver.
101+
<Display>
102+
\begin{tikzcd}
103+
1 \ar[r, "a"]
104+
&amp; 2 \ar[r, "b"]
105+
\ar[loop below, phantom, ""{coordinate, name=X} ]
106+
&amp; 3 \ar[ll, "c", rounded corners,
107+
to path={
108+
-- ([xshift=1.5ex]\tikztostart.east)
109+
|- (X)[pos=1]\tikztonodes
110+
-| ([xshift=-1.5ex]\tikztotarget.west)
111+
-- (\tikztotarget.west)
112+
}
113+
]
114+
\end{tikzcd}
115+
</Display>
116+
<P />
117+
</Alt>
118+
(In other words, this quiver is the <M>3</M>-cycle quiver, and the
119+
relations are the paths of length <M>4</M>.) The nonzero paths of length
120+
<M>2</M> are: <C>a*b</C>, <C>b*c</C>, <C>c*a</C>.
121+
<P />
122+
123+
This algebra is a Nakayama algebra, and so has finite representation type.
124+
<E>A fortiori</E>, it is syzygy-finite.
125+
</Subsection>
126+
127+
<Subsection Label="ExAlg3">
128+
<Heading>
129+
Algebra <M>3</M>
130+
</Heading>
131+
132+
The quiver and relations of this algebra are specified to &QPA; as follows.
133+
<Example><![CDATA[
134+
gap> quiv := Quiver(
135+
> 4,
136+
> [ [1,2,"a"], [2,3,"b"], [3,4,"c"], [4,1,"d"], [4,4,"e"], [1,2,"f"],
137+
> [2,3,"g"], [3,1,"h"] ]
138+
> );
139+
<quiver with 4 vertices and 8 arrows>
140+
gap> pa := PathAlgebra( Rationals, quiv );
141+
<Rationals[<quiver with 4 vertices and 8 arrows>]>
142+
gap> rels := [
143+
> pa.a * pa.g, pa.b * pa.h, pa.c * pa.e, pa.d * pa.f,
144+
> pa.e * pa.d, pa.f * pa.b, pa.g * pa.c, pa.h * pa.a,
145+
> pa.a * pa.b * pa.c * pa.d * pa.a - ( pa.f * pa.g * pa.h )^2 * pa.f,
146+
> pa.d * pa.a * pa.b * pa.c - ( pa.e )^3,
147+
> pa.c * pa.d * pa.a * pa.b * pa.c,
148+
> ( pa.h * pa.f * pa.g )^2 * pa.h
149+
> ];
150+
[ (1)*a*g, (1)*b*h, (1)*c*e, (1)*d*f, (1)*e*d, (1)*f*b, (1)*g*c,
151+
(1)*h*a, (1)*a*b*c*d*a+(-1)*f*g*h*f*g*h*f, (-1)*e^3+(1)*d*a*b*c,
152+
(1)*c*d*a*b*c, (1)*h*f*g*h*f*g*h ]
153+
]]></Example>
154+
<Alt Only="LaTeX">
155+
Here is a picture of the quiver.
156+
<Display>
157+
\begin{tikzcd}
158+
1 \ar[r, bend left, "a"] \ar[r, "f"']
159+
&amp; 2 \ar[d, bend left, "b"] \ar[d, "g"']
160+
\\
161+
4 \ar[u, bend left, "d"] \ar[loop left, "e"]
162+
&amp; 3 \ar[l, bend left, "c"] \ar[ul, "h"]
163+
\end{tikzcd}
164+
</Display>
165+
<P />
166+
</Alt>
167+
168+
The relations of this algebra are chosen so that the nonzero paths of
169+
length <M>2</M> are: <C>a*b</C>, <C>b*c</C>, <C>c*d</C>, <C>d*a</C>,
170+
<C>e*e</C>, <C>f*g</C>, <C>g*h</C> and <C>h*f</C>.
171+
</Subsection>
172+
173+
<Subsection Label="ExAlg4">
174+
<Heading>
175+
Algebra <M>4</M>
176+
</Heading>
177+
178+
The quiver and relations of this algebra are specified to &QPA; as follows.
179+
<Example><![CDATA[
180+
gap> quiv := Quiver(
181+
> 8,
182+
> [ [ 1, 1, "a" ], [ 1, 2, "b" ], [ 2, 2, "c" ], [ 2, 3, "d" ],
183+
> [ 3, 4, "e" ], [ 4, 3, "f" ], [ 3, 4, "g" ], [ 4, 5, "h" ],
184+
> [ 5, 6, "i" ], [ 6, 5, "j" ], [ 5, 7, "k" ], [ 7, 6, "l" ],
185+
> [ 6, 7, "m" ], [ 7, 8, "n" ], [ 8, 8, "o" ], [ 8, 1, "p" ] ]
186+
> );
187+
<quiver with 8 vertices and 16 arrows>
188+
gap> pa := PathAlgebra( Rationals, quiv );
189+
<Rationals[<quiver with 8 vertices and 16 arrows>]>
190+
gap> rels := [
191+
> pa.a * pa.a, pa.b * pa.d, pa.c * pa.c, pa.d * pa.g, pa.e * pa.h,
192+
> pa.f * pa.e, pa.g * pa.f, pa.h * pa.k, pa.i * pa.m, pa.j * pa.i,
193+
> pa.k * pa.n, pa.l * pa.j,
194+
> pa.m * pa.l, pa.n * pa.p, pa.o * pa.o, pa.p * pa.b,
195+
> pa.a * pa.b * pa.c * pa.d,
196+
> pa.e * pa.f * pa.g * pa.h,
197+
> pa.g * pa.h * pa.i * pa.j * pa.k,
198+
> pa.c * pa.d * pa.e - pa.d * pa.e * pa.f * pa.g,
199+
> pa.f * pa.g * pa.h * pa.i - pa.h * pa.i * pa.j * pa.k * pa.l,
200+
> pa.j * pa.k * pa.l * pa.m * pa.n - pa.m * pa.n * pa.o,
201+
> pa.o * pa.p * pa.a * pa.b - pa.p * pa.a * pa.b * pa.c
202+
> ];
203+
]]></Example>
204+
The relations of this algebra are chosen so that the nonzero paths of
205+
length <M>2</M> are: <C>a*b</C>, <C>b*c</C>, <C>c*d</C>, <C>d*e</C>,
206+
<C>e*f</C>, <C>f*g</C>, <C>g*h</C>, <C>h*i</C>, <C>i*j</C>, <C>j*k</C>,
207+
<C>k*l</C>, <C>l*m</C>, <C>m*n</C>, <C>n*o</C>, <C>o*p</C> and <C>p*a</C>.
208+
</Subsection>
209+
210+
<Subsection Label="ExAlg5">
211+
<Heading>
212+
Algebra <M>5</M>
213+
</Heading>
214+
215+
The quiver and relations of this algebra are specified to &QPA; as follows.
216+
<Example><![CDATA[
217+
gap> quiv := Quiver(
218+
> 4,
219+
> [ [ 1, 2, "a" ], [ 2, 3, "b" ], [ 3, 4, "c" ], [ 4, 1, "d" ],
220+
> [ 1, 2, "e" ], [ 2, 3, "f" ], [ 3, 1, "g" ], [ 4, 4, "h" ] ]
221+
> );
222+
<quiver with 4 vertices and 8 arrows>
223+
gap> pa := PathAlgebra( Rationals, quiv5 );
224+
<Rationals[<quiver with 4 vertices and 8 arrows>]>
225+
gap> rels := [
226+
> pa.a * pa.f, pa.b * pa.g, pa.c * pa.h, pa.d * pa.e, pa.e * pa.b,
227+
> pa.f * pa.c, pa.g * pa.a, pa.h * pa.d,
228+
> pa.b * pa.c * pa.d * pa.a * pa.b * pa.c,
229+
> pa.d * pa.a * pa.b * pa.c * pa.d * pa.a,
230+
> ( pa.h )^6,
231+
> pa.a * pa.b * pa.c * pa.d * pa.a * pa.b -
232+
> pa.e * pa.f * pa.g * pa.e * pa.f * pa.g * pa.e * pa.f,
233+
> pa.c * pa.d * pa.a * pa.b * pa.c * pa.d -
234+
> pa.g * pa.e * pa.f * pa.g * pa.e * pa.f * pa.g
235+
> ];
236+
[ (1)*a*f, (1)*b*g, (1)*c*h, (1)*d*e, (1)*e*b, (1)*f*c, (1)*g*a,
237+
(1)*h*d, (1)*b*c*d*a*b*c, (1)*d*a*b*c*d*a, (1)*h^6,
238+
(1)*a*b*c*d*a*b+(-1)*e*f*g*e*f*g*e*f,
239+
(1)*c*d*a*b*c*d+(-1)*g*e*f*g*e*f*g ]
240+
]]></Example>
241+
The relations of this algebra are chosen so that the nonzero paths of
242+
length <M>2</M> are: <C>a*b</C>, <C>b*c</C>, <C>c*d</C>, <C>d*a</C>,
243+
<C>e*f</C>, <C>f*g</C>, <C>g*e</C>, <C>h*h</C>.
244+
</Subsection>
245+
</Section>
246+
247+
<#/GAPDoc>

0 commit comments

Comments
 (0)