Insights into Mathematics | The Square Lattice via group D4 and its hypergroups | Diffusion Symmetry 5 | N J Wildberger @njwildberger | Uploaded 2 years ago | Updated 21 hours ago
Hypergroups are remarkable probabilistic/ algebraic objects that have a close connection to groups, but that allow a transformation of non-commutative problems into the commutative setting. This gives powerful new tools for harmonic analysis in situations ruled by symmetry.
Bravais lattices in 2D are important for chemists, and they come in four crystal families, although there are altogether 5 different lattices. The square lattice is particularly important also in physics and mathematics, but how do we describe its symmetry? And can the theory of hypergroups be applied?
We start with a classical view of the square lattice space group, involving both a point group D4 and a lattice normal subgroup. The finite group D4 is particularly important, arguably the second most important finite group after the symmetric group S3. We explain its character table, and then move on to the much less familiar associated class and character hypergroups.
[ERRATA: Note that on Slide 4 I made a mistake: the product of C_1 and C_2 ought to be 2C_3, not C_3. The same goes for the product C_1 and C_3, which ought to be 2C_2, and the product C_2 and C_3 which ought to be C_1. Thanks to Declan O'Neill for pointing out my error.]
Remarkable orthogonality relations for both the rows and columns of the hypergroup character table of D=D4 are a highlight of the lecture -- what we are describing here in this explicit example holds actually much more generally, as we will see later in the series.
Note that this connects strongly with our discussion of Dihedrons, see for example the Famous Math Problem 21 lectures, such as youtube.com/watch?v=lqH4BLHGsFw.
The Playlist for the Diffusion Symmetry series is at youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
************************
To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, or become a Patron at patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One additional advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.
Here are the Insights into Mathematics Playlists:
youtube.com/playlist?list=PL55C7C83781CF4316
youtube.com/playlist?list=PL3C58498718451C47
youtube.com/playlist?list=PL5A714C94D40392AB
youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
youtube.com/playlist?list=PL8403C2F0C89B1333
youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
youtube.com/playlist?list=PL6763F57A61FE6FE8
youtube.com/playlist?list=PLBF39AFBBC3FB30AF
youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
Here are the Wild Egg Maths Playlists (some available only to Members!)
youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy
youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv
youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_
youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW
youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX
youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd
youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
youtube.com/playlist?list=PLzdiPTrEWyz4IknbwXMEVwxOWS3z1Ch3C
************************
Hypergroups are remarkable probabilistic/ algebraic objects that have a close connection to groups, but that allow a transformation of non-commutative problems into the commutative setting. This gives powerful new tools for harmonic analysis in situations ruled by symmetry.
Bravais lattices in 2D are important for chemists, and they come in four crystal families, although there are altogether 5 different lattices. The square lattice is particularly important also in physics and mathematics, but how do we describe its symmetry? And can the theory of hypergroups be applied?
We start with a classical view of the square lattice space group, involving both a point group D4 and a lattice normal subgroup. The finite group D4 is particularly important, arguably the second most important finite group after the symmetric group S3. We explain its character table, and then move on to the much less familiar associated class and character hypergroups.
[ERRATA: Note that on Slide 4 I made a mistake: the product of C_1 and C_2 ought to be 2C_3, not C_3. The same goes for the product C_1 and C_3, which ought to be 2C_2, and the product C_2 and C_3 which ought to be C_1. Thanks to Declan O'Neill for pointing out my error.]
Remarkable orthogonality relations for both the rows and columns of the hypergroup character table of D=D4 are a highlight of the lecture -- what we are describing here in this explicit example holds actually much more generally, as we will see later in the series.
Note that this connects strongly with our discussion of Dihedrons, see for example the Famous Math Problem 21 lectures, such as youtube.com/watch?v=lqH4BLHGsFw.
The Playlist for the Diffusion Symmetry series is at youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
************************
To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, or become a Patron at patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One additional advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.
Here are the Insights into Mathematics Playlists:
youtube.com/playlist?list=PL55C7C83781CF4316
youtube.com/playlist?list=PL3C58498718451C47
youtube.com/playlist?list=PL5A714C94D40392AB
youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
youtube.com/playlist?list=PL8403C2F0C89B1333
youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
youtube.com/playlist?list=PL6763F57A61FE6FE8
youtube.com/playlist?list=PLBF39AFBBC3FB30AF
youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
Here are the Wild Egg Maths Playlists (some available only to Members!)
youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy
youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv
youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_
youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW
youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX
youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd
youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
youtube.com/playlist?list=PLzdiPTrEWyz4IknbwXMEVwxOWS3z1Ch3C
************************
