@@ -11,82 +11,67 @@ Demonstration: Symmetric functions
1111
1212::
1313
14- sage: SymmetricFunctions?
14+ sage: SymmetricFunctions? # not tested
1515
16- ::
16+ The ring of symmetric functions over the rational numbers ::
1717
18- sage: S = SymmetricFunctions(QQ) # The ring of symmetric functions over the rational numbers
18+ sage: S = SymmetricFunctions(QQ)
1919
20+ Typing 'objectname.<tab>' gives a lot of information about what
21+ you can do with the object::
2022
21- ::
23+ sage: S. # not tested
2224
23- sage: # Typing 'objectname.<tab>' gives a lot of information about what
24- sage: # you can do with the object
25- sage: S.
25+ The usual bases for symmetric functions::
2626
27- ::
28- 29- sage: # The usual bases for symmetric functions
3027 sage: p = S.powersum(); s = S.schur(); m = S.monomial(); h = S.homogeneous(); e = S.elementary()
3128
29+ The 'forgotten basis' is dual to the elementary basis::
3230
33- ::
34- 35- sage: # The 'forgotten basis' is dual to the elementary basis
3631 sage: f = e.dual_basis()
3732
33+ Different ways of entering symmetric functions::
3834
39- ::
40- 41- sage: # Different ways of entering symmetric functions
4235 sage: p[2,1] == p([2,1]) and p[2,1] == p(Partition([2,1]))
4336 True
4437
45- ::
38+ Changing bases ::
4639
47- sage: # Changing bases
4840 sage: p(s[2,1])
4941 1/3*p[1, 1, 1] - 1/3*p[3]
5042
51- ::
43+ Sums of different bases are automatically converted to a single basis ::
5244
53- sage: # Sums of different bases are automatically converted to a single basis
5445 sage: h[3] + s[3] + e[3] + p[3]
5546 2*h[1, 1, 1] - 5*h[2, 1] + 6*h[3]
5647
57- ::
48+ Littlewood-Richardson coefficients are relatively fast ::
5849
59- sage: # Littlewood-Richardson coefficients are relatively fast
6050 sage: timeit('s[10]^4')
61- 5 loops, best of 3:..
51+ 5 loops, best of 3:...
6252
63- ::
53+ Changing bases ::
6454
65- sage: # Changing bases
66- sage: time h(s[10]^4);
55+ sage: h(s[10]^4);
6756 h[10, 10, 10, 10]
68- Time: CPU 1.07 s, Wall: 1.08 s
6957
70- ::
58+ We get an arbitrary symmetric function to demonstrate some functionality ::
7159
72- sage: # We get an arbitrary symmetric function to demonstrate some functionality
73- sage: foo = h.an_element()
60+ sage: foo = 1/2*h([]) + 3*h([1, 1, 1]) + 2*h([2, 1, 1])
7461 sage: foo
7562 1/2*h[] + 3*h[1, 1, 1] + 2*h[2, 1, 1]
7663
77- ::
64+ The omega involution ::
7865
79- sage: foo.omega() # The omega involution
66+ sage: foo.omega()
8067 1/2*h[] + 3*h[1, 1, 1] + 2*h[1, 1, 1, 1] - 2*h[2, 1, 1]
8168
82- ::
83- 8469 sage: e(foo.omega())
8570 1/2*e[] + 3*e[1, 1, 1] + 2*e[2, 1, 1]
8671
87- ::
72+ The Hall scalar product ::
8873
89- sage: foo.scalar(s[3,1]) # The Hall scalar product
74+ sage: foo.scalar(s[3,1])
9075 4
9176
9277::
@@ -99,60 +84,44 @@ Demonstration: Symmetric functions
9984 sage: foo.skew_by(e[2,1])
10085 9*h[] + 10*h[1]
10186
102- ::
87+ We can define skew partition directly ::
10388
104- sage: # We can define skew partition directly
10589 sage: mu = Partition([3,2])/Partition([2,1])
10690 sage: mu
107- [[3, 2], [2, 1]]
108- 109- ::
91+ [3, 2] / [2, 1]
11092
11193 sage: s(mu)
11294 s[1, 1] + s[2]
11395
114- ::
96+ We can expand a symmetric function in monomials ::
11597
116- sage: # We can expand a symmetric function in monomials
11798 sage: s(mu).expand(3)
11899 x0^2 + 2*x0*x1 + x1^2 + 2*x0*x2 + 2*x1*x2 + x2^2
119100
120- ::
101+ Or we can choose our alphabet ::
121102
122- sage: # Or we can choose our alphabet
123103 sage: s(mu).expand(3,alphabet=['a','b','c'])
124104 a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2
125105
126- ::
127- 128106 sage: mu = Partition([32,18,16,4,1])/Partition([14,3,2,1])
129107 sage: la = Partition([33,19,17,4,1])/Partition([15,4,3,1])
130108
131- 132- ::
133- 134109 sage: (s(la) - s(mu)).is_schur_positive()
135110 True
136111
137- ::
138- 139112 sage: foo.kronecker_product(foo)
140113 1/4*h[] + 54*h[1, 1, 1] + 20*h[1, 1, 1, 1] + 8*h[2, 1, 1]
141114
142- ::
143- 144115 sage: foo.plethysm(h[3])
145116 1/2*h[] + 3*h[3, 3, 3] + 2*h[4, 3, 3, 2] - 2*h[5, 3, 3, 1] + 2*h[6, 3, 3]
146117
147118::
148119
149- sage: foo.inner_plethysm?
150- 120+ sage: foo.inner_plethysm? # not tested
151121
152- ::
122+ The transition matrix from the Schur basis to the power basis
123+ Try s.transition_matrix? for more information::
153124
154- sage: # The transition matrix from the Schur basis to the power basis
155- sage: # Try s.transition_matrix? for more information
156125 sage: s.transition_matrix(m,5)
157126 [1 1 1 1 1 1 1]
158127 [0 1 1 2 2 3 4]
@@ -162,9 +131,8 @@ Demonstration: Symmetric functions
162131 [0 0 0 0 0 1 4]
163132 [0 0 0 0 0 0 1]
164133
165- ::
134+ The sum of degree 6 Schur functions whose first part is even ::
166135
167- sage: # The sum of degree 6 Schur functions whose first part is even
168136 sage: foo = sum([s[mu] for mu in Partitions(6) if mu[0]%2 == 0])
169137 sage: foo
170138 s[2, 1, 1, 1, 1] + s[2, 2, 1, 1] + s[2, 2, 2] + s[4, 1, 1] + s[4, 2] + s[6]
@@ -175,16 +143,13 @@ Demonstration: Symmetric functions
175143 ....: r""" Remove the last part from a partition """
176144 ....: return Partition(mu[:-1])
177145
146+ We can apply this map to all the partitions appearing in 'foo'::
178147
179- ::
180- 181- sage: # We can apply this map to all the partitions appearing in 'foo'
182148 sage: foo.map_support(remove_last_part)
183149 s[] + s[2, 1, 1, 1] + s[2, 2] + s[2, 2, 1] + s[4] + s[4, 1]
184150
185- ::
151+ Warning! This gives different results depending on the basis in which foo is expressed ::
186152
187- sage: # Warning! This gives different results depending on the basis in which foo is expressed
188153 sage: h(foo).map_support(remove_last_part)
189154 3*h[] + h[2, 1, 1, 1] + h[2, 2] - 2*h[2, 2, 1] - 2*h[3, 1, 1] + 2*h[3, 2] - 2*h[4] + 4*h[4, 1] - 4*h[5]
190155
@@ -193,62 +158,47 @@ Demonstration: Symmetric functions
193158 sage: foo.map_support(remove_last_part) == h(foo).map_support(remove_last_part)
194159 False
195160
196- ::
161+ We can easily get specific coefficients ::
197162
198- sage: # We can easily get specific coefficients
199163 sage: foo.coefficient([4,2])
200164 1
201165
202- ::
203- 204- sage: # There are many forms of symmetric functions in sage.
205- sage: # They do not (yet) all appear under 'SymmetricFunctions'
206- sage: # These are the ~H[X;q,t] often called the 'modified Macdonald polynomials'
207- sage: Ht = MacdonaldPolynomialsHt(QQ)
208- 166+ There are many forms of symmetric functions in sage.
167+ These are the `~H[X;q,t] ` often called the 'modified Macdonald polynomials'::
209168
169+ sage: S = SymmetricFunctions(FractionField(QQ['q','t']))
170+ sage: Ht = S.macdonald().Ht(); Ht
171+ Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Ht basis
172+ sage: p = S.powersum(); s = S.schur(); m = S.monomial(); h = S.homogeneous(); e = S.elementary();
210173::
211174
212175 sage: s(Ht([3,2]))
213- Traceback (most recent call last):
214- ...
215- TypeError
176+ q^4*t^2*s[1, 1, 1, 1, 1] + (q^4*t+q^3*t^2+q^3*t+q^2*t^2)*s[2, 1, 1, 1] + (q^4+q^3*t+q^2*t^2+q^2*t+q*t^2)*s[2, 2, 1] + (q^3*t+q^3+2*q^2*t+q*t^2+q*t)*s[3, 1, 1] + (q^3+q^2*t+q^2+q*t+t^2)*s[3, 2] + (q^2+q*t+q+t)*s[4, 1] + s[5]
216177
217178::
218179
219180 sage: Ht.base_ring()
220181 Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field
221182
222- ::
223- 224- sage: S.base_ring()
225- Rational Field
226- 227183::
228184
229185 sage: q
230186 Traceback (most recent call last):
231187 ...
232188 NameError: name 'q' is not defined
233189
234- ::
190+ The following is a shortcut notation (based on Magma).
191+ It defines R to be the polynomial ring in the variables
192+ 'q' and 't' over the rational numbers, and makes these variables
193+ available for use::
235194
236- sage: # The following is a shortcut notation (based on Magma).
237- sage: # It defines R to be the polynomial ring in the variables
238- sage: # 'q' and 't' over the rational numbers, and makes these variables
239- sage: # available for use
240195 sage: R.<q,t> = Frac(ZZ['q','t'])
241196
242- 243- ::
244- 245197 sage: S = SymmetricFunctions(R)
246- 247- ::
248- 249198 sage: p = S.powersum(); s = S.schur(); m = S.monomial(); h = S.homogeneous(); e = S.elementary();
250- sage: Ht = MacdonaldPolynomialsHt(R)
251199
200+ sage: Ht = S.macdonald().Ht(); Ht
201+ Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Integer Ring in the Macdonald Ht basis
252202
253203::
254204
@@ -258,80 +208,62 @@ Demonstration: Symmetric functions
258208::
259209
260210 sage: latex(_)
261- q^{4} t^{2}s_{1,1,1,1,1} + \left(q^{4} t + q^{3} t^{2} + q^{3} t + q^{2} t^{2}\right)s_{2,1,1,1} + \left(q^{4} + q^{3} t + q^{2} t^{2} + q^{2} t + q t^{2}\right)s_{2,2,1} + \left(q^{3} t + q^{3} + 2 q^{2} t + q t^{2} + q t\right)s_{3,1,1} + \left(q^{3} + q^{2} t + q^{2} + q t + t^{2}\right)s_{3,2} + \left(q^{2} + q t + q + t\right)s_{4,1} + s_{5}
211+ q^{4} t^{2}s_{1,1,1,1,1} + \left(q^{4} t + q^{3} t^{2} + q^{3} t + q^{2} t^{2}\right)s_{2,1,1,1} + \left(q^{4} + q^{3} t + q^{2} t^{2} + q^{2} t + q t^{2}\right)s_{2,2,1} + \left(q^{3} t + q^{3} + 2 q^{2} t + q t^{2} + q t\right)s_{3,1,1} + \left(q^{3} + q^{2} t + q^{2} + q t + t^{2}\right)s_{3,2} + \left(q^{2} + q t + q + t\right)s_{4,1} + s_{5}
262212
263213::
264214
265215 sage: s(Ht([3,2])).coefficient([2,1,1,1]).subs({q:q^(-1), t:t^(-1)}) *q^5 * t^5
266216 q^3*t^3 + q^2*t^4 + q^2*t^3 + q*t^4
267217
268- ::
218+ We can also create the ring of Macdonald Polynomials
219+ using different parameters::
269220
270- sage: # We can also create the ring of Macdonald Polynomials
271- sage: # using different parameters
272221 sage: A.<a,b> = QQ[]
273- sage: P = MacdonaldPolynomialsP (FractionField(A),a,b )
274- sage: sa = SymmetricFunctions(FractionField(A)) .schur()
275- 222+ sage: S = SymmetricFunctions (FractionField(A))
223+ sage: sa = S .schur()
224+ sage: P = S.macdonald(a,b).P()
276225
277226::
278227
279228 sage: sa(P[2,1])
280229 ((a*b-b^2+a-b)/(-a*b^2+1))*s[1, 1, 1] + s[2, 1]
281230
282- ::
283- 284- sage: # Press <tab> after the following to see the different
285- sage: # variants of Macdonald polynomials in sage
286- sage: MacdonaldPolynomials
287- Traceback (most recent call last):
288- ...
289- NameError: name 'MacdonaldPolynomials' is not defined
231+ Press <tab> after the following to see the different
232+ variants of Macdonald polynomials in sage
290233
291- ::
234+ sage: Sym = SymmetricFunctions(FractionField(QQ['q',t']))
235+ sage: J = Sym.macdonald()
236+ sage: J.TAB # not tested
292237
293- sage: # Press <tab> after the following to see the different
294- sage: # variants of Jack polynomials in sage
295- sage: JackPolynomials
238+ Press <tab> after the following to see the different variants of Jack polynomials in sage::
296239
240+ sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
241+ sage: J = Sym.jack()
242+ sage: J.TAB # not tested
297243
298- ::
244+ Press <tab> after the following to see the different
245+ variants of Hall-Littlewood polynomials in sage
299246
300- sage: # Press <tab> after the following to see the different
301- sage: # variants of Hall-Littlewood polynomials in sage
247+ sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
248+ sage: H = Sym.hall_littlewood()
302249 sage: HallLittlewood
303250
251+ Now some examples with `k `-Schur functions::
304252
305- ::
306- 307- sage: ks2 = kSchurFunctions(R,2,t=R(t))
308- sage: s = SymmetricFunctions(R).schur()
309- 253+ sage: Sym = SymmetricFunctions(QQ['t'])
254+ sage: KB = Sym.kBoundedSubspace(2)
255+ sage: ks2 = KB.kschur()
256+ sage: s = Sym.schur()
310257
311258::
312259
313260 sage: s(ks2[2,2,1])
314261 s[2, 2, 1] + t*s[3, 1, 1] + (t^2+t)*s[3, 2] + (t^3+t^2)*s[4, 1] + t^4*s[5]
315262
316- ::
317- 318263 sage: ks2(s[1])
319264 ks2[1]
320265
321- ::
322- 323266 sage: ks2(s[3])
324267 Traceback (most recent call last):
325268 ...
326- ValueError: s[3] is not in the space spanned by k-Schur Functions at level 2 over Multivariate Polynomial Ring in q, t over Rational Field.
327- 328- ::
329- 330- sage: # Warning: Not well supported yet!
331- sage: SchubertPolynomialRing
332- 333- 334- ::
335- 336- sage: # Warning: Not well supported yet!
337- sage: LLT
269+ ValueError: s[3] is not in the image
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