DemazureProductWithInverse from SignedPermutation

The Demazure product π⁻¹ ⋆ π in the hyperoctahedral group.

SignedPermutation → SignedPermutation

FindStat's Mp00260: π⁻¹ ⋆ π, the inverse on the left. Not injective: it is a surjection onto the involutions. The group is generated by the adjacent transpositions and the sign change of the largest value.

Called as CombinatorialMap(x, "DemazureProductWithInverse"). The head's page has its examples.

Live

Explore

Each element of SignedPermutation at a size, and its image, applied in your browser; sizes past 6 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on SignedPermutation. Its distribution over the images of SignedPermutation at each size is computed, and compared with the 18 statistics on SignedPermutation: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic NegativeEntryCountPositiveEntryCountTypeBInversionCountNegativeSumPairCountFlagMajorIndexFlagDescentFlagExcedanceFlagWeakExcedancePositiveBeforeLargerNegativePairCountIncreasingPositivePairCountPositiveBeforeNegativeCountRightDescentsSignedPermutationOrderSignedReflectionLengthFlagDenertIndexSignedSortingIndexSignedAlignmentCountStasinskiVollLength Sizes 1 to

FindStat Mp00260

Demazure product with inverse

Matched by value: our map agrees with FindStat's image on each of the 1200 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

Graded mapas FindStat records them, checked by FindStat's contributors.

This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on signed permutations.

Object	Image
[1]	[1]
[-1]	[-1]
[1,2]	[1,2]
[2,1]	[2,1]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00260 has the rest.