powerTraces -- Compute the traces of powers of a matrix

Description

This is what you do:

i1 : ea = exteriorExtension(3, 6, QQ);

i2 : A = ea.ad(first ea.bases#1);

              55       55
o2 : Matrix QQ   <-- QQ

i3 : ea.powerTraces A

     +-----+-----+----+
o3 = |power|trace|rank|
     +-----+-----+----+
     |1    |0    |20  |
     +-----+-----+----+
     |2    |0    |1   |
     +-----+-----+----+

The list outputs the trace of each power until the power ranks stabilize, or until the power is equal to the dimension of the algebra. These are invariants of the element x in ad(x).

i4 : ea = exteriorExtension(3, 9, QQ);

i5 : E = ea.appendage

o5 = E

o5 : PolynomialRing, 9 skew commutative variable(s)

i6 : A = ea.ad(E_0*E_1*E_2 + E_3*E_4*E_5 + E_6*E_7*E_8);

              248       248
o6 : Matrix QQ    <-- QQ

i7 : ea.powerTraces A

     +-----+-------------------------------------------------------------+----+
o7 = |power|trace                                                        |rank|
     +-----+-------------------------------------------------------------+----+
     |1    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |2    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |3    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |4    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |5    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |6    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |7    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |8    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |9    |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |10   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |11   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |12   |4536                                                         |168 |
     +-----+-------------------------------------------------------------+----+
     |13   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |14   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |15   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |16   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |17   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |18   |-117936                                                      |168 |
     +-----+-------------------------------------------------------------+----+
     |19   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |20   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |21   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |22   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |23   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |24   |3188808                                                      |168 |
     +-----+-------------------------------------------------------------+----+
     |25   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |26   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |27   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |28   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |29   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |30   |-86093280                                                    |168 |
     +-----+-------------------------------------------------------------+----+
     |31   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |32   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |33   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |34   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |35   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |36   |2324523096                                                   |168 |
     +-----+-------------------------------------------------------------+----+
     |37   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |38   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |39   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |40   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |41   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |42   |-62762119056                                                 |168 |
     +-----+-------------------------------------------------------------+----+
     |43   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |44   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |45   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |46   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |47   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |48   |1694577219048                                                |168 |
     +-----+-------------------------------------------------------------+----+
     |49   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |50   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |51   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |52   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |53   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |54   |-45753584909760                                              |168 |
     +-----+-------------------------------------------------------------+----+
     |55   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |56   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |57   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |58   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |59   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |60   |1235346792568056                                             |168 |
     +-----+-------------------------------------------------------------+----+
     |61   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |62   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |63   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |64   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |65   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |66   |-33354363399332976                                           |168 |
     +-----+-------------------------------------------------------------+----+
     |67   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |68   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |69   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |70   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |71   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |72   |900567811781994888                                           |168 |
     +-----+-------------------------------------------------------------+----+
     |73   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |74   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |75   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |76   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |77   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |78   |-24315330918113857440                                        |168 |
     +-----+-------------------------------------------------------------+----+
     |79   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |80   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |81   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |82   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |83   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |84   |656513934789074155416                                        |168 |
     +-----+-------------------------------------------------------------+----+
     |85   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |86   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |87   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |88   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |89   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |90   |-17725876239305002191696                                     |168 |
     +-----+-------------------------------------------------------------+----+
     |91   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |92   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |93   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |94   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |95   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |96   |478598658461235059180328                                     |168 |
     +-----+-------------------------------------------------------------+----+
     |97   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |98   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |99   |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |100  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |101  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |102  |-12922163778453346597864320                                  |168 |
     +-----+-------------------------------------------------------------+----+
     |103  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |104  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |105  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |106  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |107  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |108  |348898422018240358142341176                                  |168 |
     +-----+-------------------------------------------------------------+----+
     |109  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |110  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |111  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |112  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |113  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |114  |-9420257394492489669843207216                                |168 |
     +-----+-------------------------------------------------------------+----+
     |115  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |116  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |117  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |118  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |119  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |120  |254346949651297221085766599368                               |168 |
     +-----+-------------------------------------------------------------+----+
     |121  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |122  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |123  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |124  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |125  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |126  |-6867367640585024969315698178400                             |168 |
     +-----+-------------------------------------------------------------+----+
     |127  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |128  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |129  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |130  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |131  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |132  |185418926295795674171523850821336                            |168 |
     +-----+-------------------------------------------------------------+----+
     |133  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |134  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |135  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |136  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |137  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |138  |-5006311009986483202631143972171536                          |168 |
     +-----+-------------------------------------------------------------+----+
     |139  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |140  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |141  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |142  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |143  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |144  |135170397269635046471040887248636008                         |168 |
     +-----+-------------------------------------------------------------+----+
     |145  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |146  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |147  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |148  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |149  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |150  |-3649600726280146254718103955713167680                       |168 |
     +-----+-------------------------------------------------------------+----+
     |151  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |152  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |153  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |154  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |155  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |156  |98539219609563948877388806804255531896                       |168 |
     +-----+-------------------------------------------------------------+----+
     |157  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |158  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |159  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |160  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |161  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |162  |-2660558929458226619689497783714899356656                    |168 |
     +-----+-------------------------------------------------------------+----+
     |163  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |164  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |165  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |166  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |167  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |168  |71835091095372118731616440160302282634248                    |168 |
     +-----+-------------------------------------------------------------+----+
     |169  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |170  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |171  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |172  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |173  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |174  |-1939547459575047205753643884328161631120160                 |168 |
     +-----+-------------------------------------------------------------+----+
     |175  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |176  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |177  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |178  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |179  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |180  |52367781408526274555348384876860364040248856                 |168 |
     +-----+-------------------------------------------------------------+----+
     |181  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |182  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |183  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |184  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |185  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |186  |-1413930098030209412994406391675229829086714576              |168 |
     +-----+-------------------------------------------------------------+----+
     |187  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |188  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |189  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |190  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |191  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |192  |38176112646815654150848972575231205385341298088              |168 |
     +-----+-------------------------------------------------------------+----+
     |193  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |194  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |195  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |196  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |197  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |198  |-1030755041464022662072922259531242545404215043840           |168 |
     +-----+-------------------------------------------------------------+----+
     |199  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |200  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |201  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |202  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |203  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |204  |27830386119528611875968901007343548725913806188216           |168 |
     +-----+-------------------------------------------------------------+----+
     |205  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |206  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |207  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |208  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |209  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |210  |-751420425227272520651160327198275815599672767077296         |168 |
     +-----+-------------------------------------------------------------+----+
     |211  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |212  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |213  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |214  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |215  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |216  |20288351481136358057581328834353447021191164711091528        |168 |
     +-----+-------------------------------------------------------------+----+
     |217  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |218  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |219  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |220  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |221  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |222  |-547785489990681667554695878527543069572161447199466720      |168 |
     +-----+-------------------------------------------------------------+----+
     |223  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |224  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |225  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |226  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |227  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |228  |14790208229748405023976788720243662878448359074385605976     |168 |
     +-----+-------------------------------------------------------------+----+
     |229  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |230  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |231  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |232  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |233  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |234  |-399335622203206935647373295446578897718105695008411356816   |168 |
     +-----+-------------------------------------------------------------+----+
     |235  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |236  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |237  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |238  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |239  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |240  |10782061799486587262479078977057630238388853765227106638568  |168 |
     +-----+-------------------------------------------------------------+----+
     |241  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |242  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |243  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |244  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |245  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |246  |-291115668586137856086935132380556016436499051661131879236800|168 |
     +-----+-------------------------------------------------------------+----+
     |247  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+
     |248  |0                                                            |168 |
     +-----+-------------------------------------------------------------+----+

The list could be quite long, as this example shows.

For the programmer

The object powerTraces is a symbol.

The source of this document is in ExteriorExtensions.m2:619:0.