Typography of Lie groups, Lie algebras

This is a note for myself as I review my notes on finite groups, Lie groups (both the classic infinite ones and the finite simple groups of Lie type), Lie algebras, and specifically the typography for them.

First, no one agrees completely, and everyone's conventions is slightly different. What I mean by this: although everyone agrees, e.g., that the A_n family of Lie algebras refers to the same thing, some people use serif font for the "A", others use bold, some italicize, others do not, etc.

I'm inclined to follow Robert Wilson's conventions from his book The Finite Simple Groups (2009).

Families of Groups

Families of Simple Lie groups: use upright, teletype font for the family and "math italics" for the subscript if it's a variable, e.g., $\mathtt{A}_{n}$, $\mathtt{B}_{5}$, $\mathtt{C}_{m}$, $\mathtt{D}_{n^{2}}$, $\mathtt{E}_{8}$

Exceptional Finite Simple Groups of Lie Type: don't treat the formatting as special, so, e.g., the Steinberg groups would be ${}^{2}A_{n}(q^{2})$, ${}^{2}D_{n}(q^{2})$, ${}^{2}E_{6}(q^{2})$, ${}^{3}D_{4}(q^{3})$.

Sporadic simple group: these should be made upright, e.g., the Suzuki group is $\mathrm{Suz}$, the Matthieu groups look like $\mathrm{M}_{11}$, the Conway groups $\mathrm{Co}_{1}$ and $\mathrm{Co}_{2}$, and so on. BUT the exception to this rule is that the Monster group is written $\mathbb{M}$ and the Baby Monster $\mathbb{B}$.

Alternating, Cyclic, Symmetric group. These are just written as $A_{n}$, $C_{n}$, or $S_{n}$. The dihedral group, too, is $D_{n}$.

Classical Lie Groups: Here there is a double standard. For classical Lie groups over the reals or complex numbers, we write something of the form $\mathrm{GL}(n, \mathbb{F})$, $\mathrm{SL}(n, \mathbb{F})$, $\mathrm{U}(n, \mathbb{F})$, $\mathrm{SU}(n, \mathbb{F})$, $\mathrm{O}(n, \mathbb{F})$, $\mathrm{SO}(n, \mathbb{F})$, $\mathrm{Sp}(n)=\mathrm{USp}(n)$ for the compact Symplectic group, $\mathrm{Sp}(2n,\mathbb{F})$ for the generic Symplectic group.

The finite groups corresponding to these guys are written a little differently in my notes: the $n$ parameter is pulled out as a subscript, because frequently we write $q$ instead of $\mathbb{F}_{q}$ for finite fields...and then looking at $\mathrm{SL}(8,9)$ is far more confusing than $\mathrm{SL}_{8}(9)$. Thus we have $\mathrm{GL}_{n}(q)$, and so on.

Projective classical groups: the projective classical groups are prefixed by a "P", not a blackboard bold $\mathbb{P}$. E.g., $\mathrm{PSL}_{2}(7)$. At present, the projective orthogonal group wikipedia page seems to agree with this convention.

Operations

For finite groups: the Atlas of finite groups seems to have set the standard conventions for finite groups, Wilson changes them slightly. We'll find $G = N{:}H$ for the semidirect product $G = N \rtimes H = H\ltimes N$. Also $A\mathop{{}^{\textstyle .}}\nolimits B = A{\,}^{\textstyle .} B$ for a non-split extension with quotient $B$ and normal subgroup $A$, but no subgroup $B$. And $A{.}B$ is an unspecified extension.

Lie algebras. Writing notes on paper, for a given Lie group $G$, I write $\mathrm{Lie}(G)$ as its Lie algebra. (It turns out to be a functor...neat!) If I have to write Fraktur by hand, I approximate it using Pappus's caligraphy tutorial.

Huygens's Appendix to De ratiociniis in aleae ludo

Quoting from an English translation of Hugens's appendix to De ratiociniis in aleae ludo [or, On Reasoning in Games of Dice], here are the five problems given:

Problem 1. A and B play together with a pair of Dice upon this Condition, That A shall win if he throws 6, and B if he throws 7; and A is to take one Throw first, and then B two Throws together, then A to take two Throws together, and so on both of them the same, till one wins. The Question is, What Proportion their Chances bear to one another? Answ. As 10355 to 12276.

Problem 2. THREE Gamesters, A, B, and C, taking 12 Counters, 4 of which are White, and 8 black, play upon these Terms: That the first of them that shall blindfold choose a white Counter shall win; and A shall have the first Choice, B the second, and C the third; and then A to begin again, and so on in their turns. What is the Proportion of their Chances?

Problem 3. A lays with B, that out of 40 cards, i.e. 10 of each different Sort, he will draw 4, so as to have one of every Sort. And the Proportion of his Chance to that of B, is found to be as 1000 to 8139.

Problem 4. HAVING chosen 12 Counters as before, 8 black and 4 white, A lays with B that he will blindfold take 7 out of them, among which there shall be 3 black ones. Quaere [Lat., "Search out"], What is the Proportion of their Chances?

Problem 5. A and B taking 12 Pieces of Money each, play with 3 Dice on this Condition, That if the Number 11 is thrown, A shall give B one Piece, but if 14 shall be thrown, then B shall give one to A; and he shall win the Game that first gets all the Pieces of Money. And the Proportion of A's Chance to B's is found to be, as 244,140,625 to 282,429,536,481.

This is also known as the "gambler's ruin" problem. Pascal first posed this problem to Fermat (and it was communicated to Huygens via Carcavi in a letter dated September 28, 1656). To be clear: A keeps tossing the dice until the game is over.

For more about the history and context of these problems, see Anders Hald's A History of Probability and Statistics and Their Applications Before 1750 pp.74–78.

Bernoulli's Problems from Ars Conjectandi

Part III of Bernoulli's Ars Conjectandi discusses 23 problems. Here I'd like to collect them.

Problem 1. Someone with two tokens, one white and one black, hidden in an urn, offers a prize to three people, A, B, and C, on the condition that whoever draws out the white token will win the prize and if all fail to do so none will win the prize. A draws first and replaces the token, B second, C third. What are their lots?

Problem 2. Other things remaining as posited in the previous problem, if the organizer of the game of chance, wanting to deny to himself any right to the prize, tells the others to divide the prize among themselves if none of them should choose the white token, what then will be their lots?

Problem 3. Six players A, B, C, D, E, F, by order of the organizer, who favors the later players more than the earlier, attempt a game of chance. The first two, A and B, begin to play separately. The one who wins plays with the third, C, and whichever of them comes out on top competes with D, and so forth until the last player F, so that the one who emerges victorious from the last match will win the prize. Let it be supposed that each pair compete between themselves with equal lot, that is, that neither of the players has a stronger expectation of winning than the other. What are the players' lots?

Problem 4. Other things assumed to be as before, if we imagine that there is not an equal lot in any game, but that any player competing with the second player after himself has two times as many cases for winning as for losing and with the third player four times as many cases, with the fourth player eight times as many cases, and so forth—excepting only the first two players, whom we assume to compete in an equal context—then it is asked whether, in this case, all six players acquire an equal right to the offered prize, since the twice smaller expectations of the preceding proposition are compensated for the double ratio of cases.

Note that Bernoulli's notion of probability is frequentist, and the probability of an event $A$ is $\Pr(A) = N_{A}/N$ the number of outcomes $N_{A}$ which include $A$ divided by the total number of outcomes $N$. Bernoulli calls $N_{A}$ and $N$ the "number of cases".

Problem 5. A contends with B that from 40 playing cards, that is, 10 of each suit, he will choose 4 cards such that there is one of each suit. What is the ratio of their lots?

This is from Huygens's appendix to his De ratiociniis in aleae ludo, problem number three. The trick is to think in terms of "success" and "failure".

Problem 6. Taking 12 tokens, 4 white and 8 black, A contends with B that blindfolded he will choose 7 tokens from these, among which there will be 3 white ones. What is the ratio of the lots of A to the lot of B?

This is from Huygens's appendix again, problem number four. The trick is to think in terms of "success" and "failure". Bernoulli notes the ambiguity in Huygens's phrasing of the problem, and calculates both (1) the probability of obtaining exactly 3 white tokens, and (2) the probability of obtaining at least 3 white tokens.

Problem 7. Some number of players A, B, C, etc., draw cards in order and alternately from a hand of playing cards, of which one is a face card and the rest are nonface cards, on the condition that the person who draws the face card wins. Let A draw first, B second, C third, and so forth to the last, after which A proceeds to take the following card, and so forth until the end of the game. What is the ratio of their lots?

Problem 8. With other things as before, if in the hand of cards there are several face cards and if the person is judged to win who draws the first face card, then what is the ratio of lots?

Problem 9. With other things posited as before, if the players agree with each other that the one who draws more face cards will win and that if two or more players draw an equal number of face cards, they will divide the stake equally among them, while the rest who draw a smaller number of face cards will get nothing, then what is the ratio of their lots?

Problem 10.Four players A, B, C, D, having made among themselves the same agreement as in the preceding problem, play with 36 cards of which 16 are face cards and distribute to each player the cards alternately in order. It happens, however, that when 23 cards have been distributed, A has received by lot 4 face cards, B has 3, C has 2, D has 1, so that there remain 13 cards among which there are 6 face cards. The fourth player D (who is next in order to receive a card), seeing that almost all hope of his winning has vanished, wishes to sell his right to one of the others. How much should he sell it for and what are the expectations of the individual players?

Problem 11. It is proposed, with six throws of a die, to throw its six faces, each one once, so that none of the faces is repeated. What is the expectation of doing this?

Problem 12. It is proposed, with six throws of a die, to throw the six faces in order, on the first throw 1 point, on the second 2 points, on the third 3, etc. What is the expectation of achieving this?

Problem 13. Three players (A, B, C) each having written in front of himself the first six numerals, play with a die alternately, on the condition that whoever throws any number of points deletes that number from his written numbers, or, if he does not have it any more, the following player proceeds to throw. This continues until someone first deletes all his written numerals. It happens, however, that after this game has been played for a while, A still has 2 numbers before him, B has 4, and C has 3, and it is A's turn to throw. What are their lots?

Problem 14. Two players, A and B, throwing a die on a gaming tray, agree between themselves that each one will get as many throws as the number thrown on the die and that the one who throws the most points altogether will take the stake. If, however, it happens that they both get the same number of points, then they will also equally divide the stake between themselves. Afterwards, however, one of the players, B, tiring of the game, chooses instead to take a certain number of points in place of the uncertain throw of the die and tot take 12 points as his share. A assents. Which of the two players has the greater hope of winning and by how much?

Problem 15. Other things remaining as before, player B requests to be conceded as his share of points the square of the number of points on the first throw. Now what is the ratio of lots?

Problem 16.The valuation of lot in the game called Cinq et Neuf.

In France, Denmark, Sweden, Belgium, lower Germany, and neighboring regions, a kind of game is played that they call Cinq et Neuf; it is played by two people, A and B, with two dice; one of them, A, receives unending turns. These are the conditions of play: if A on the first turn throws a 3 or 11 or any pair (un doublet, ein Pasch), that is, two ones, two twos, threes, etc., then A wins. If A throws a 5 or 9, the other player, B, wins. If A throws any other number of points, namely 4, 6, 7, 8, or 10, then neither of the players wins, but the game continues until a 5 or 9 is thrown, in which case B is the winner, or until exactly the same number of points is thrown again as was thrown on the first throw, in which case A wins. The condition concerning the throwing of a 3 or 11 or of any pair does not aid A except on the first throw. On these assumptions what is the ratio of lots?

Problem 17. The valuation of the lot in a certain other kind of game of chance.

I remember once seeing here at the time of the weekly market a certain peddler who was explaining the following kind of game in the marketplace and attracting to it those passing by. There was a circular disk made quite level, mounting upwards for a little while toward the center. The border was surrounded by 32 contiguous and equal small pockets or openings that were marked into four distinct classes or series by numbers written four times in order from I up to VIII. A dice box hung perpendicularly over the middle of the disk. The one about to make a trial of fortune dropped through the cavity of the dice box four little balls to be received by an equal number of compartments in the circumference of the disk. He received as a prize what the sum of the numbers in the compartments indicated, which might be larger or smaller depending on the sum, as the prizes in the table indicate. For each throw of the balls, the player was to pay four coins. What is the player's expectation?

Points	Coins	Cases
4	120	1
5	100	16
6	30	52
7	24	128
8	18	245
9	10	416
10	6	664
11	6	976
12	6	1369
13	5	1776
14	3	2204
15	3	2560
16	3	2893
17	2	3088
18	2	3184
19	3	3088
20	3	2893
21	3	2560
22	3	2204
23	4	1776
24	4	1369
25	6	976
26	8	664
27	12	416
28	16	245
29	24	128
30	25	52
31	32	16
32	180	1

Problem 18. On the card game called in the vernacular Trijaques.

Very common among the Germans is the kind of game called Trijaques, which has an affinity to the French game Brelan. From a deck of cards 24 are taken (with the rest set aside), six from each suit, the nines, tens, jacks, queens, kings, and aces, which herafter will be referred to by their initials NTJQKA. These cards have the following priorities: in first place is the ace, followed by the king, and then the queen, jack, and ten, but higher than all these are the nines, together with the jack of clubs (which, accordingly, we will consider to be a nine, so that there are 5 nines and only 3 jacks). The preeminence of the nines, which, rather like the cards in the Spanish game Hombre called Matadors, they call robbers, murderers, or assassins, consisting in the fact that they are judged to be of any suit or number one pleases. Thus 2 nines with an ace, or 1 nine with 2 aces, make 3 aces or three-of-a-kind (un Tricon) of aces. One, 2, or 3 nines, joined with 3, 2, or 1 kings make four-of-a-kind of kings. One or two nines together with three or two cards of the same suit make four of that suit, for example, four hearts, spades, clubs, etc., the sort of combination of cards customarily called a flush (ein Fluss), which is valued beyond the number of points. The nine or the ace is worth 11 points while the other ranks of cards are each worth 10 points.

This is the way the game is played: To each of the players in order two cards are dealt. After these are privately examined, the first player is free to stake any sum of money. Another player who wishes to join the play stakes an equal amount or even adds something extra if he pleases, which the first player must match if he does not wish to lose his stake. After this is completed, each player who remains in the game is dealt two more cards face up on the table, so that part of each player's hand is visible to the other players and part is hidden. Then they begin to wager money again, alternately increasing the stake as before and challenging the others to bid or concede. In the end each player shows his hand to the others and the one who has the best hand takes the whole stake. Now four-of-a-kind is better than a flush, and a flush is better than three-of-a-kind, but 4 nines are best of all. In the other cases of three-of-a-kind and four-of-a-kind, the order follows the order of the cards, and in flushes the order follows th enumber of points. So, for example, 3 or 4 aces beats 3 or 4 kings, and a flush of 43 points beats one of 42 points. If none of the players has three- or four-of-a-kind or a flush, then the one who has the most points of the same suit takes the stake. If two hands are equal in every way, for example, if 2 players have three-of-a-kind or four-of-a-kind of the same rank, or if they each have a flush containing an equal number of points, then the one who is closer to the dealer or who received his cards first wins.

This seems like a "toy model" of Texas hold 'em.

Problem 19. Suppose any kind of game in which the organizer of the game or cashier (the banker of the game) has some advantage consisting in the fact that the number of cases in which he wins is a little larger than those in which he loses and furthermore in that the number of cases in which the banker remains in office for the following game is slightly larger than the number in which the office is transferred to another player. How great is the advantage of the banker?

Bernoulli considers a throw of dice where the number of cases in which the banker wins to the number of cases in which he loses is in ratio of p to q (greater to less). Also the number of cases in wihch the banker retains his position for the following throw to the number of cases in which the position is transferred to a fellow player to be in ratio of m to n (again of greater to less).

Problem 20. The evaluation of the lot in the card game commonly called Capriludium or Bockspiel.

Two or more players use playing cards. One competes against the others and performs the office of banker (he has the bank), after he has shuffled the cards and divided the pack into as many hands as there are players including himself. Then the individual players buy individual hands for a price laid down, leaving the last to the banker. Finally, the banker turns the hands over and reveals their bottom cards, besides which no others are considered. Once this has been done, the banker is expected to pay those whose cards are of a higher rank than his as much as each one laid out for the chance. But those to whom fell a card of a lower rank than or equal to the card of the banker suffers the loss of their stake, ceding this rather to the payment of the victorious banker, who also continues to perform his office as long as he has defeated at least one of his opponents, nor does he leave the position of banker unless he has been bested by absolutely all his fellow players.

Bernoulli considers problem 19 as a simpler "toy model" of problem 20.

Problem 21. On the game of Basset.

This game is most celebrated because of the innumerable tumults and tragedies it once excited on all sides here and there, especially in Italy and in France because of which it was proscribed from these regions not long after and prohibited under severe penalty. At the time when the play of this game flourished most greatly in the hall of the King of France, D. Salvator (Sauveur), a French mathematician and preceptor of the Dauphin, subjected the expectations of the players to calculation, and published some tables of them in the Parisian Journal des Scavans for February 1679, with a brief synopsis. From this journal, I will review those aspects of the nature and constitution of the game that seem necessary for the examination of the Tables and for the eliciting of the calculations that the author suppressed.

After the banker has taken the whole deck of playing cards and shuffled it, each of the players before him at the table lays out a card of any rank (taken from elsewhere) together with a chosen sum of money. Then the banker turns over his deck of cards, revealing the bottom card, and deals out the cards successively two at a time in order until he has exhausted the pack. When he is dealing each of the pairs of cards, the first or preceding card favors the banker and the latter or following card is advantage of the opponent. For instance, if the first card turned up is a king, then the banker takes whatever has been laid on kings. If, however, the king comes up second, the banker must give to each of the players in turn as much as they have laid on kings. Up to this point of the rules of the game, no one has any advantage over another. But in addition the following rules should be noted:

1. If both cards are of the same weight and rank as a displayed card (which they call doublets, and we will call twins), in which case the profit and loss ought to balance each other out, only the banker benefits and takes whatever is laid on the cards of that rank.

2. Any player, even in the middle of a hand, has the renewed right to lay new money on any card. When this happens it may occur that only 1, 2, 3, or all 4 of the cards of that rank remain in the pack—which will markedly affect his lot. But it should be observed that a pair of which he has seen either card is judged to be of no value with respect to the rank that has been bought; and indeed if the latter card of this sort should agree in rank, not only does it profit nothing for the player, being as it were premature [trop jeune], but it also puts an end to the round, with respect to the option of betting on another card. If, on the other hand, that rank should appear on the first card of the following pair, it has a diminished value [c'est une face] and to the banker goes a profit of only two-thirds of the stake.

3. The prior card of the first pair, moreover, since there may be some suspicion that it may have been seen on the part of the banker, is always of diminished value, and can only give the winner two-third of the prize.

4. Since, when there is only one card left of a rank for which the players contest, there is no opportunity for twin cards, in which the advantage of the banker consists it is ruled in favor of the banker that the last card of all, which ought otherwise to benefit the player, should be considered null.

Salvator's article seems to be available online, well, it's been scanned by Google Books pp. 43 et seq.

Problem 22. There is a certain kind of game in which the number of all the cases is a, the number of some of them is b, and the number of the rest is $a-b=c$. Titius buys single throws of a single die by paying single coins to Caius. However many times he throws one of the b cases, he receives m coins back from Caius, but however many times he throws one of the c cases, he receives nothing. If, however, he throws one of the c cases continuously for n times, Caius is obliged to return all of his n coins. What are the lots of Titius and Caius?

Problem 23. On the game of blind dice [blinde Wurffel].

By this name are called those six dice adopted by most of our peddlers, which are cube-shaped like ordinary dice, but appear blank, with individual dice marked with points on only one face, one die with one point, another with two, a third with three, up to the sixth, which has one of its faces marked with six points. Thus, all together, only $1+2+3+4+5+6=21$ points are found. Imposters display this sort of dice in the marketplace for cheating the populace, together with prizes for all the numbers of points from 1 up to 21 in amounts that can be seen in the adjoined table. Then a person who wishes to risk his fortune, having paid coin to the peddler, throws the dice onto the tray, and if he has thrown some numbers of points, takes the prize assigned for that number. If, however, no points fall to him, he loses his coin.

This assumed, anyone who wants to investigate the lot of these players should note the following:

1. That the number of all the cases in six dice of this sort, no different from ordinary dice, is 46,656, namely as many as the sixth power of six.

2. That the number of cases that carry absolutely no points is equal to the sixth power of five (15625) since on any die there are five blank faces of which any one can be combined with any of the five blank faces of another die, and of these combinations again any can be combined with the five blank faces of the third die, so that the number of preceding cases is always quintupled.

3. That any number of points may be produced by one, two, or more dice. If it is produced by one, on the other five dice there will be no point showing, so that, since there are five blank faces on each die, the number of cases in which that happens will be 3125 (the fifth power of 5). If the number of points is produced by two dice, on the remaining four there will be no points, so that noe, the number of cases in which this occurs will be 625 (the fourth power of 5).

4. The same number of points can be produced not only frequently by more or fewer dice, but also sometimes by the same number of dice in several ways. Thus 12 points can be produced by three ways and two ways by four dice (on three dice the points can be 1,5,6; or 2,4,6; or 3,4,5; and on four dice, the points can be 1,2,3,6; or 1,2,4,5).

Problem 24. Assuming again the situation in the previous problem, if the master of the game contracts with the player that he wishes to be bound to restore to the latter all his coins if no points fall to him on five throws in a row, what then will be the lot of each?